The Collected Papers of Alfred Young 1873–1940 9781487575625

Table of contents :
Contents
Foreword
Alfred Young, 1873-1940
Collected Papers. 1899: The Irreducible Concomitants of any Number of Binary Quartics
1901: The Invariant Syzygies of Lowest Degree for any number of Quartics
1901: On Quantitative Substitutional Analysis
1902: On Quantitative Substitutional Analysis (Second Paper)
1902: On Quadratic Invariant Types
1903: The Expansion of the nth Power of a Determinant
1903: The Maximum Order of an Irreducible Covariant of a System of Binary Forms
1904: On Covariant Types of Binary n-ics
1905: Perpetuant syzygies
1905: On Certain Classes of Syzygies
1908: On Relations among Perpetuants
1914: On Binary Forms
1920: The Electromagnetic Properties of Coils of Certain Forms
1924: Ternary Perpetuants
1926: The Linear Invariants of Ten Quaternary Quadrics
1928: On Quantitative Substitutional Analysis (Third Paper)
1928: On Quantitative Substitutional Analysis
1930: On Quantitative Substitutional Analysis (Fourth Paper)
1930: On Quantitative Substitutional Analysis (Fifth Paper)
1931: On Quantitative Substitutional Analysis (Sixth Paper)
1933: Binary Forms with a Vanishing Covariant of Weight Four or Five
1933: Note on Transvectants
1933: Some Generating Functions
1934: On Quantitative Substitutional Analysis (Seventh Paper)
1934: On Quantitative Substitutional Analysis (Eighth Paper)
1935: The Application of Substitutional Analysis to Invariants
1952: On Quantitative Substitutional Analysts (Ninth Paper)

Citation preview

This volume includes all the published papers of Alfred Young (1873-1940), who made outstanding contributions to the algebra of invariants and the theory of groups, together with a biographical sketch published after Young's death and a foreword by Professor G. de B. Robinson of the University of Toronto. It will be of interest to algebraists, combinatorialists, theoretical physicists, and students of the corresponding disciplines.

MATHEMATICAL EXPOSITIONS

EDITORIAL BOARD

H.S.M. COXETER, G.F.D. DUFF, D.A.S. FRASER, G. de B. ROBINSON (Secretary), P.G. ROONEY VOLUMES PUBLISHED

1. The Foundations of Geometry. G. de B. Robinson 2. Non-Euclidean Geometry. H.S.M. Coxeter 3. The Theory of Potential and Spherical Harmonics. W.J. Sternberg and T.L. Smith 4. The Variational Principles of Mechanics. Cornelius Lanczos 5. Tensor Calculus. J.L. Synge and A.E. Schild (out of print) 6. The Theory of Functions of a Real Variable. R.L. Jeffery ( out of print) 7. General Topology. Waclaw Sierpinski; translated by C. Cecilia Krieger (out of print) 8. Bernstein Polynomials. G.G. Lorentz ( out of print) 9. Partial Differential Equations. G.F.D. Duff 10. Variational Methods for Eigenvalue Problems. S.H. Gould 11. Differential Geometry. Erwin Kreyszig ( out of print) 12. Representation Theory of the Symmetric Group. G. de B. Robinson 13. Geometry of Complex Numbers. Hans Schwerdtfeger 14. Rings and Radicals. N.J . Divinsky 15. Connectivity in Graphs. W.T. Tutte 16. Introduction to Differential Geometry and Riemannian Geometry. Erwin Kreyszig 17. Mathematical Theory of Dislocations and Fracture. R.W. Lardner 18. n-gons. Friedrich Bachmann and Eckart Schmidt; translated by Cyril W.L. Gamer 19. Weighing Evidence in Language and Literature: A Statistical Approach. Barron Brainerd 20. Rudiments of Plane Affine Geometry. P. Scherk and R. Lingenberg 21. The Collected Papers of Alfred Young, 1873-1940

MATHEMATICAL EXPOSITIONS NO . 21

The Collected Papers of Alfred Young 1873-1940 F.R.S., Sc.D., LL.D. Fellow of Clare College, Cambridge Canon of Chelmsford Rural Dean of Belchamp Rector of Bird brook, Essex 1910-1940

University of Toronto Press Toronto and Buffalo

© University of Toronto Press 1977 Toronto and Buffalo Printed in Canada Reprinted in 2018

Canadian Cataloguing in Publication Data

Young, Alfred, 1873-1940. The collected papers of Alfred Young, 1873-1940 (Mathematical expositions ; 21

ISSN 0076-5333)

ISBN 0-8020-2267-7

ISBN 978-1-4875-7272-3 (paper)

1. Mathematics - Collected works. Gilbert de B., 1906II. Series. QA3.Y69

510'. 8

1.

Robinson ,

c77-001375-9

This book has been published during the Sesquicentennial year of the University of Toronto

Contents

Foreword, by G. de B. Robinson

vii

Alfred Young, 1873-1940, by H.W. Turnbull (JLMS 16 (1941): 194-207)

xv

COLLECTED PAPERS

1899 *The irreducible concomitants of any number of binary quartics (PLMS 30: 290-307) 1901 *The invariant syzygies of lowest degree for any number of quartics (PLMS 32 : 384-404) *On quantitative substitutional analysis 1 (PLMS 33 : 97-146) 1902 *On quantitative substitutional analysis 11 (PLMS 34: 361-397) On quadratic invariant types (Mess. Math. 32 : 57-59) 1903 The expansion of the nth power of a determinant (Mess. Math. 33 : 113-116) The maximum order of an irreducible covariant of a system of binary forms (Proc. Roy. Soc. 72 : 399-400) 1904 On covariant types of binary n-ics (PLMS 1: 202-208) 1905 Perpetuant syzygies (PLMS 2: 221-265), with P.W. Wood On certain classes of syzygies (PLMS 3: 62-82) 1908 *On relations among perpetuants (Trans. Camb. Phil. Soc. 20: 66-73) 1914 On binary forms (PLMS 13 : 441-495) 1920 The electromagnetic properties of coils of certain forms (PLMS 18 : 280-290) 1924 *Ternary perpetuants (PLMS 22 : 171-200) 1926 *The linear invariants of ten quaternary quadrics (Trans. Camb. Phil. Soc. 23: 265-301), with H.W. Turnbull 1928 *On quantitative substitutional analysis m (PLMS 28: 255-292) *On quantitative substitutional analysis (JLMS 3: 14-19) 1930 *On quantitative substitutional analysis JV (PLMS 31 : 253-272) *On quantitative substitutional analysis v (PLMS 31 : 273-288)

3 21 42 92 129 132 136 138 145 190 211 219 274 285 315 352 390 396 416

vi 1931 *On quantitative substitutional analysis VI (PLMS 34: 196-230) 1933 Binary fonns with a vanishing covariant of weight four or five (JLMS 8 : 182-187) Note on transvectants (JLMS 8 : 187-188) *Some generating functions (PLMS 35: 425-444) 1934 *On quantitative substitutional analysis VII (PLMS 36: 304-368) *On quantitative substitutional analysis VIII (PLMS 37: 441-495) 1935 *The application of substitutional analysis to invariants (Phil. Trans. Roy. Soc. 234 : 79-114) 1952 *On quantitative substitutional analysis IX (PLMS 54: 219-253) *These papers involve group theory.

432 467 472 474 494 559 614 650

Foreword

Invariant theory and its applications were popular in the 19th century, but the work of Frobenius, Burnside, and Schur shifted the emphasis to group theory, in particular to the representation theory of symmetric groups, a subject which is of importance in quantum mechanics and modern combinatorics. Later developments have aroused so much interest that a symposium was held in Oberwolfach in I 975. At this symper sium, on 'Combinatorics: Young Tableaux and Combinatorics in Symmetric Group Representation,' the need was expressed to go back to the pioneering work in the field , particularly that of Alfred Young. It was urged upon me, as Young's sole Ph.D. student in Cambridge, to bring out a volume of his collected papers - which are not now readily available in many libraries. It was suggested also that I might include some samples of his correspondence with me. I am grateful that the original publishers of the papers looked upon the suggestion with favour and granted the necessary permissions for reproduction. Alfred Young's life and contribution to mathematical knowledge are well summarized in the biography by H.W. Turnbull published in the Journal of the London Mathematical Society in 1941 following his sudden death on 15 December 1940. It serves as an appropriate introduction to the present volume. A few months after Young's death, on 31 March 1941, his wife wrote me a charming letter, saying in part: I was so overwhelmed with the sudden passing of my dear one, he was out with me on the Wednesday afternoon visiting in the parish. He seemed as usual, when suddenly after tea he was taken with a pain in his side . The doctor had him rushed off to the hospital, operated upon him but he never really came around and passed away on Sunday morning. I know it was what he would have wished - to die at his post. Mr. Hall of King's College, Cambridge has kindly promised to look through his unpublished papers; he came over and took them all ...

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Subsequently, H.W. Richmond, also of King's College, and Turnbull went through the papers after Philip Hall. The large box of papers was eventually sent to me at the University of Toronto in 1948, where they are now housed in the Robarts Library and are available for consultation by anyone interested. My introduction to the ninth paper on quantitative substitutional analysis (see pp. 650ff.) provides a summary of the contents noted in an examination by W.T. Sharp and me. It is worth mentioning here that Alfred Young appreciated the need to interpret his representation theory of Sn to apply to a subgroup H c Sn. Some day someone will continue where he left off and achieve the success which I still believe is possible. In spite of his love of abstract ideas, Alfred Young was a very practical man. The rectory at Birdbrook was large and very beautiful. Turnbull refers to his small electrical engine used to pump water. During the war years 1914-18, Young worked on the paper published in 1920 (pp. 274-84), and in the box of his papers we found two patents. Following the 1976 Strasbourg sequel to the Oberwolfach symposium, I went to the Patent Office in London and obtained the specifications. Professor G.R. Siemon, Chairman of the Department of Electrical Engineering at the University of Toronto, examined these, commenting as follows : Patent Specification 1 13,679 is entitled 'An Apparatus for the purpose of the Direct Conversion of the Energy of Motion of a Dielectric into Electric Energy, and Conversely for the Conversion of Electric Energy ( of the right frequency) into the Motion of Mass.' This apparatus, patented in 1918, is similar in many ways to the magnetohydrodynamic generators currently under development for efficient conversion of thermo energy to electric energy. The invention converts mechanical energy in a moving fluid such as water or gas to high-frequency electric energy without the use of moving parts. Exploitation of the basic principle of this invention has had to wait on the development of superconducting coils to produce intense magnetic fields. While present-day magnetohydrodynamic generators produce either direct current or line-frequency alternating current, Young set out to produce highfrequency electric currents predominantly for use in the emerging technology of wireless telegraphy. The Patent Specification 127,488 is entitled 'A Machine for the Generation of Electric Currents also Applicable as a Motor.' The patent, granted in 1919, is also for an apparatus to produce high-frequency alternating currents. It is one of a family of devices in which the mutual inductance between coils is varied by virtue of their relative motion and the varying inductance is used in a tuned circuit to produce high-frequency current. Similar devices existed at that time but were restricted

ix to the production of low-frequency current by the inclusion of iron cores. The advantage claimed for this device was that it could operate at high frequencies because it used air-cored coils. To the best of my knowledge, neither of these inventions was exploited during the period of validity of the patent. Both specifications display a sound knowledge of electrical principles. There is no question that the devices would work . Very few inventions, however, reach the stage of engineering practicality and these are among the many that have failed to achieve industrial application. Alfred Young appears to have been a genuine natural philosopher with added ingenuity.

After visiting the Patent Office, our son-in-law drove my wife and me up to Birdbrook, where I took the accompanying photographs. Mrs Young lived until 1950 and she and her husband are remembered in a stained glass window in the ancient church to which they both contributed so much over so many years. Young's group theoretical ideas were developed in the long series of papers entitled 'On Quantitative Substitutional Analysis' (QSA), but his other applications of group theory to invariants are also of interest. Here we include all of Young's papers, whether on these or other topics, so it seemed desirable to distinguish those involving group theory by adding an asterisk before the title in the table of contents. Not included in the volume is Young's only book, Algebra of Invariants (Cambridge University Press, 1903), co-authored with J.H. Grace, which Turnbull describes ecstatically. The first six chapters are devoted to the 19th century approach using differential operators, determinants, and generating functions, and the remainder is largely of geometrical interest (Grace was a geometer) except for the last chapter in which Young describes his process of 'symmetrization. ' In QSA 1 (pp. 42-9 l) Young derives the fundamental equation 1 = ~AaTa and applies his ideas to proving theorems of Gordan, Capelli, and Peano. In QSA 11 (pp. 92-128) he calculates Aa and studies the properties of the PN. The paper concludes with an application to binary forms. Burnside refereed QSA I and 11, and Young told me that he advised him to read the works of Frobenius and Schur. But Young knew no German and it was not till QSA m (pp. 352-89) appeared in 1928 that he further pursued his group-theoretic ideas. In the introduction to this, one of his most important papers, he relates his work to that of Frobenius and proceeds to develop the idempotents and their properties in a masterly fashion. Young's work had attracted Weyl's attention (see Grnppentheorie und Quantenmechaniks [2nd ed., Hirzel, Leipzig, 1931] ), and suddenly he was famous.

X

Church of St. Augustine of Canterbury, Birdbrook, Essex

xi

Grave of Dr. and Mrs. Alfred Young, Birdbrook, Essex

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In QSA 1v, v, and v 1 (pp. 396-466) Young pursues and largely completes his contribution to group theory. He had been continuing his study of Frobenius and Schur and in QSA v he applied his method to obtain the representation theory of the hyperoctahedral group. In QSA VI he derives the fundamental rule for writing down the orthogonal representation of Sn, starting with the hook-representation which I had dealt with in 1931 in my thesis (see Representation Theory of the Symmetric Group [University of Toronto Press, 1961] ). In the remaining papers Young returns to his first love - invariant theory - and applies his symmetrizer to organize the field. In QSA v 11 and v m (pp. 494-6 I 3) he comes to realize the significance of the sequence in which standard tableaux can be written. This is proving increasingly important in recent developments of the subject. In his 1935 paper he further relates his ideas to work of Frobenius and Schur. My introduction to QSA IX (pp. 650-84) concludes this summary of Young's work. As Turnbull so ably expresses it, Young's contributions had already been appreciated in 1941, and it is my hope that this volume will make them more accessible to another generation. The papers are presented as they appeared originally, including the pagination of the journal at the top of the page for ease in following the many cross-references (the pagination of the present volume is at the foot of the pages). For the same reason the year of publication is placed in a prominent position in the table of contents. The papers are reproduced in facsimile, which precluded any adjustment of the text except for the correction of minor typographical errors noted on copies sent to me by Young. Unfortunately the reproduction presented a problem in places where the type was weak; I ask the reader to bear with these imperfections. The following excerpts from two letters of Alfred Young relating to my first graduate student H.H. Ferns, whose thesis was published in the Transactions of the Royal Society of Canada, 111 (1934), 35-60, may be of interest. June 19, 1933 I have been trying to get to grips with Ferns' dissertation and suddenly it seemed to me that a good deal of what he was after was very intimately connected with something I had included in my last paper Q.S. A. vm which I sent in to the L.M.S. at the beginning of February last, and heard from Watson about a fortnight or so ago that it had been accepted for publication. As it is unlikely to get into print until some time next year, I thought I had better copy out the two sections of that paper

xiii

from my MS, (I had it typed for the L.M.S. to have the MS by me), and let you have them , in case they may be of some assistance to you or Fems. Of course the matter is approached from a different point of view ; and with another purpose in prospect. But I think these two sections are understandable of themselves - they lead on from Q .S. A. VI (with probably the exception ofpp. 4,5 , i.e. Theorem xv in the enclosed MS - this has nothing to do with the matter at present in discussion of Fems' work: - but of the greatest importance for my object - invariant algebra - it was put in here to avoid a gap in what I sent you) .. . On later thoughts I have also copied out and enclose Section VII of Q.S. A. VIII, which is closely connected with the subject ; and is an illustration of Section VI. I should be immensely interested to see a geometrical interpretation of the double group matrix. There is a uniqueness about the irreducible representations, that might correspond to some very fundamental geometric property. There is probably some general simple form of the irreducible double group matrix applicable to all cases which it would be interesting to discover: - but I have had no time to go further into that than the discussion enclosed. I should imagine that Fems is a man who should be encouraged as much as possible, he appears to possess that infinite capacity for taking pains which is the bedrock of success. And also to have the capacity of imagination which should lead him to tum his labours in the right direction. August 26, 1933 ... Many thanks for letting me see Fems' work, which I have posted separately. His ideas should lead him on to quite a general theory, both interesting and useful. I hope he will not feel I have cut any ground from under his feet in that extract of Q.S. A. vm I sent you ; but there is ample room for him to develope his own way of looking at the algebra, and the geometry is a wide field which he has begun well to explore. I am in great hopes that my MS may help him to generalise his own work, and to cut down the mass of algebraic detail with which it is rather embarassed at present. We have had the most wonderful summer I ever remember, the harvest is practically finished and appears to be excellent in every way. Many thanks for asking us to visit you in Canada, I should love to do so. But my wife is very averse to travelling, even to crossing the channel, so I much regret that it is a most unlikely event.

It is appropriate to conclude this Foreword by referring to the recent revival of interest in invariant theory (see C. Procesi, 'The Invariant Theory of n X n Matrices,' Advances in Mathematics, l 9 [ 1976], 30681). It is not so much the geometrical aspects which are being emphasized but the general algebraic structure which was so well described in the Algebra of Invariants and to which Young returned in his later years.

xiv Since this Foreword was written the proceedings of the Strasbourg 'Table Ronde' have appeared in print (Combinatoire et Representations du Groupe Symetrique [Springer-Verlag, Lecture Notes in Mathematics No. 579, 1977] ), and it is interesting to read the many papers presented there in the present context. It is expected that the next volume in the Mathematical Expositions Series will be one by T.V. Narayana dealing with the application of Young's ideas in statistics. G. de B. Robinson Toronto, 1977

ALFRED YOUNG 1873-1940 H. W. TURNBULL

Alfred Young was bom at Birchfield, Farnworth, near Widnes, Lancashire, on 16 April 1873. He died after a short illness on Sunday, 15 December 1940. He was the youngest son of Edward Young, a prosperous Liverpool merchant and a Justice of the Peace for the county. His father married twice and had a large family, eleven living to grow up. The two youngest sons of the two branches of the family rose to scientific distinction: Sydney Young, of the elder family, became a distinguished chemist of Owen's College, Manchester, Uni.ersity College, Bristol, and finally, for many years, of Trinity College, Dublin. He was elected Fellow of the Royal Society in his thirty-sixth year and died in 193i. Alfred, who was fifteen years his junior, was elected Fellow in 1934, at the age of sixty, in recognition of his mathematical contributions to the algebra of invariants and the theory of groups, a work to which he had devoted over ten years of academic life followed by thirty years of leisure during his duties as rector of a country parish. Recognition of his remarkable powers came late but swiftly; he was admitted to the Fellowship in the year when his name first came up for election. In 1879 the family moved to Bournemouth, and in due course the younger brothers ,vent to school and later to a tutor, under whom Alfred suffered for his brain power, being the only boy considered worth keeping in . Next, he went to Monkton Combe School, near Bath, and there again his unusual mathematical ability was recognised. Thence he gained a scholarship at Clare College, Cambridge, where he matriculated in 1892. At Clare he formed his life-long friendship with G. H. A. l\'ilson, another distinguished mathematician, who eventually became Master of the College. Young was a good oarsman and rowed in the Junior Trial Eights as a freshman and in the Scratch Fours of 1893. His college friends still remember him as a shy, clever lad with a great humility of spirit which so marked him in his youth and indeed throughout his life. Early in his third year at college his interest in research began, and his enthusiasm doubtless diverted him from the subjects laid down for the Tripos examina-

xvi

tion, for which he was prepared by the celebrated coach, Webb of St. John's College. In 1895 he graduated as tenth \\'rangler, his friend Wilson being placed fifth. It was a brilliant year; Bromwich was Senior Wrangler, Grace and Whittaker were bracketed second, and thereafter followed Hopkinson, Godfrey, and Maclaurin twelfth Wrangler, to whom one of the three Smith's Prizes was subsequently awarded. Young, who according to Grace was the most original man of his year, would probably have occupied a higher place in the list had he directed his attention to the examination schedule; but in turning to his research, as Wilson tells us, undoubtedly he chose the better path. In the following year Young was placed in the Second Class of the Mathematical Tripos, Part II. From 1901 to 1905 he lectured at Selwyn College, Cambridge, but resigned that appointment shortly after his election to a Fellowship at his own college, where he was also Bursar until 1910. His work received recognition when he was approved for the degree of Sc.D. at Cambridge in 1908. Young had always intended to take Holy Orders, but it was not until 1908 that he was ordained, when he accepted a curacy at Christ Church, Blacklands, Hastings. Two years later he was presented by his College to the living of Birdbrook, a village of Essex about twenty-five miles east of Cambridge and on the borders of Suffolk. There he lived and worked for thirty years, quietly and faithfully performing the duties of a parish priest, beloved by his congregation, who respected and wondered at his great scholastic gifts so modestly set forth. He was ahvays a welcome visitor in their homes, and he conducted the services in the parish church with dignity and sincerity, and was an excellent preacher. He readily undertook the responsibilities of his calling further afield, was appointed in 1923 to be Rural Dean of Belchamp, and in 1929 Chaplain to the Bishop of Colchester. As recently as November 1940 he was installed as Honorary Canon of Chelmsford Cathedral. In 1926 he accepted an invitation from the University of Cambridge to give a course of lectures on Higher Algebra, and this he continued to do for several years during the Lent, and occasionally the May, Term. In 1931 he was awarded the Honorary LL.D. Degree of the University of St. Andrews. In 1907 Alfred Young married Edith Clara, daughter of :Mr. Edward Wilson, of Sheffield, by whom he is survived. There ,vere no children of the marriage. Their home was a typical country rectory, set in an old world garden full of colour and of great charm, where a warm welcome awaited a visitor from Cambridge or elsewhere, young or old, who sought out in this secluded corner of Essex a master of abstract algebra, and found more than a mathematician, a friend. After a thirty-mile bicycle ride (these were the days before the motor bus had become ubiquitous)

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the shade and peace of the rectory garden on a summer day were greatly refreshing. " He was charming to us ", an undergraduate wrote after such a visit, "and I remember how delighted I was with the Rectory, and how he told us of the excellence of the beer brewed with the water of his pond " . He was a practical mechanic, and had a device by which all the water in the house was pumped up with the help of a little motor engine which was run round to the pump each day. He had also successfully set up a small electric light plant to supply the house. His expert knowledge of its working seemed odd amid those rustic surroundings, till one recalled his interests in the more practical mathematics besides the theory of groups, and the paper he had published on the electromagnetic properties of coils. He was very methodical in planning his duties and his leisure ; he was a willing correspondent and would take great pains to answer the mathematical queries of his friends, sharing ,vith them liberally his mvn abundant thoughts on algebra, invariants and geometry. He could take up or lay aside and again , after sewral weeks, resume a formidable piece of algebraic eomputation, without apparently losing the threads of the arguments and with the utmost composure. Every year he and his wife would regularly take their holiday shortly after Easter at a South Coast resort. and every Tuesday they would go over to Cambridge and thus keep in touch with their eniversity friends , a practice which doubtless started when Young returned to the lecture room at the call of the University. His quiet determination and his unhurried devotion to the things of the mind and of the spirit were very impressive. Of such might Whittier have been thinking when he wrote: "And let our ordered lives confess, The beauty of Thy peace " . Y onng was intellectually alert to the end of his life, and during his last year he was constantly working at his ninth memoir, to which he attached great importance. It was nearly finished and lay on his desk awaiting the final touches. During the last few days of illness he asked his doctor whether he could hope to live to finish his work. l\ly friend W. L. Edge has supplied a picture of the lecture course when Young resumed his teaching at Cambridge : "I remember (who could forget?) very well my experiences of attending his first lectures. This was only a course of one lecture a week for one term ; you can see for yourself how much he got through . .. . Doubtless it is all standard work to you, but it will be interesting to see how the old warrior entered the lists again and what he considered should be given to his first hearers. I went along on 19 January 1926, in my third year, just two terms before my Tripos, to Clare . . . there were eleven of us and I was the only undergraduate who ventured. Others in the class were, I think, Cooper, now

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at Belfast; Broadbent, now at Greenwich; L. H. Thomas, who got a Smith's Prize and a Trinity Fellowship and went to America; Dirac certainly, and F. P. White, the only 1\1.A. And I remember the tall clerical figure entering the room, and his surprise at so large an audience, and shaking hands with White with obvious pleasure. And so to linear transformations and Aronhold's symbolic notation . . . . At the end of his last lcctme in )larch, Young said that he was so pleased that people had turned up that he would lecture again in the following term. And he and I were both surprised, and I very embarrassed. when no other member of the cl,\ss but myself showed up in April. It was my Tripos term, but I was not going to miss his lectures ! . . . One lecture fell during the General Strike, and no preparation of room, blackboard, chalk or anything had been made by the college. So Young sat down beside me and wrote out the notes with his own hand ''. Young had a quiet humour. "I remember an occasion", \\Tites \Yilson, ·' when he said to me with a grave face: I have lost all sense of personal security'. It appeared that the maidservant at his Cambridge lodgings had used some of his manuscripts to light the fire rather than waste clean paper for that purpose". Young \\Tote his first mathematical paper in 1899 and continued to -write and to publish for over forty years. · With the exception of his work on electromagnetism in 1918, every paper was devoted to one theme, the algt•bra of groups. It began with the algebraic theory of invariants, a subject which was first explicitly started a hundred years ago, in 1841, by Boole, and then developed by Cayley, Salmon and Sylvester, and later by )facmahon and Elliott. It provided the analytical aspect of geometrical projection and of those properties of a figure which remain unchanged for any such projection. This led, first of all, to the discovery of algebraic forms which were invariant for the corresponding linear transformations, and then to the search for the basic set of forms, out of which all other invariants of a given system of ground forms could be constructed. Such a form is the binary n-ic

f (n) ! = r=O 1 2r r ar xn-•x and the study resolved itself into the theory of annihilators, that is, of certain differential operators linearly composed of terms such as

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At an early stage it was supposed that, for binary forms higher than the quartic, the invariant theory was essentially different from that for lower forms. Indeed, in his second .Memoir on Quantics (1855), Cayley had stated his conclusion that whereas the number of different irreducible invariants and covariants for a quartic was finite, this was no longer true of the quintic. But this surmise was upset in 1869 when Gordan startled the mathematical world by proving the finiteness of such systems for a binary form of any order. Gordan followed this up with the publication of his Programm at Erlangen in 1875 which widened the scope of the finiteness to all systems of binary forms. Finally, the theory was extended to all higher types of form by Hilbert in 1890. The influence of Gordan's work was apparent in the successful use of generating functions by Sylvester and l\Iadlahon, who made important advances both in detailed systems and in the general theory. A friendly rivalry sprang up between the English mathematicians and the Continental algebraists, Gordan with his followers, the former following the non-symbolic method, as it is called, and the latter the symbolic. The work of the English school is ably expounded by Elliott in his Algebra of Quantics, which brings us to the beginning of the twentieth century. The development by the symbolic method, which went on in Germany, grew out of certain hyperdeterminants invented by Cayley. In this branch of the theory the coefficient ar ·in the form f was regarded as a numerical multiple of anf/oxr:-r ox{ or, let us say, or--r i}{f, where o1 = o/ox1, o2 = ojox2• All such forms together with all their invariants and covariants were expressed in terms of the Oi, o2 belonging to the variables x 1 , x2 , and of analogous symbols for any further cogredient variables. Clebsch and Aronhold perfected the technique of these symbols. Thereupon Clebsch and Gordan used them systematically for invariant theory with conspicuous success. They proved that every rational integral invariant of binary forms could be expressed as a polynomial aggregate of symbols o2/ox 1 oy2 -o2/ox 2 oy1 (the hyperdeterminants of Cayley) and of y1 (o/ox 1 )+y2 (o/ox 2 ), the polar operators (the First Fundamental Theorem); that all invariantive properties could be deduced by means of certain specified determinantal identities which left these characteristic hyperdeterminantal and polar structures unimpaired (the Second Fundamental Theorem); that every such invariant could be expressed rationally and integrally in terms of a finite number of in,ariants (Gordan's Theorem, 1869); and they established an important expansion (the Clebsch-Gordan series) which enables one to deal with forms involving many sets of variables by means of forms in fewer variables and their polars. These symbolic advances, together with the generating functions and

xx

perpetuants of )lacl\fahon, opened up a wide fidd of enquiry, and it was this into which Young entered, in company with his friend J. H. Grace, soon after and possibly even before they took their degrees. Their interest was first aroused in 1895 by reading :\Ieyer's newly published Bericht uber den gegenwartigen Stand der lnvariantentheorie, which opened up a vast world of algebra and gave them their first ideas of modern mathematics. They came to grips \\ith the symbolic method, which fascinated them, and at once began to make important contributions to a subject hitherto very little known in England. \Yith the publication of their treatise, the Algebra of Invariants, in 1903, a new era dawned for the teaching and progress of Higher Algebra. This excellent book, with its fine display of algebraic technique and geometrical insight, had a considerable influence on the younger geometers and algebraists at Cambridge, particularly in the decade preceding the last war. Both authors were masters of their subject, Grace as a geometer and Young as an algebraist. The pages of the book have a deceptively simple appearance, owing to the extraordinary compactness of the symbolic notation, where, for example, an in,ariant a 0 a 4 -4a1 a 3 +3a 22 of a binary quartic appears in the guise 1-(ab)t. The book ends \\ith four lively appendices, bringing the latest results to the notice of the reader-and one almost expects to see a stop press column on the last page! From the outset Young·s rapid and skilful handling of symbolic algebra bore all the signs of genius. Grace likens him to Ramanujan, not only for what each achieved but for what each ignored. Young at once found his own solutions for the complete systems of the binary octavic and septimic forms: and, of his ingenious device for treating the octavic as the symbolic square of a quartic, Grace says "I could not have thought of that in fifty years". Young began his research in algebra by solving the problem of binary quanic types-the invariant theory of any number of quartics; a·nd it was through the practice of the symbolic methods upon such problems that he was led to his first great discovery, which he called Quantitative Substitutional Analysi8. He was dealing with functions of a finite number of variables; and these variables always occurred in sets, let us say a, b, c, ... , each of which could be manipulated as a single vector, or as a column of a determinant. In the course of the work innumerable varieties of alternative expressions were produced, many of which only differed among themselves by sign, or else by derangements of these vectors, in much the same way as a three-rowed determinant, ja1 b2 c3 1= ti, assumes two values ±li and six alphabetical forms, when the letters are

xxi

permuted without disturbing the suffix sequence. His functions were, of course, usually much more complicated than single determinants; nevertheless the determinant provided the clue to a general theory which comprehends a.II the details of the algebra. Now functions such as these are evidently closely connected with the theory of finite groups, and it became clear to Young that sets of functions, which at first sight were quite distinct, but which on examination proved to belong to the same group, could be dealt with by a, single prescription if only the properties of that particular group were thoroughly known. Young therefore set himself to extricate the group properties of these functions, and accordingly he expressed the functions as well as the relations between such functions by means of a new kind of symbolic operator depending at once on a substitution group. This operator, which consisted of two main ingredients Ni and P 1, can best be explained by a simple example: thus, if /(a, b, c) is a function of three variables, then j(a, b, c)+f(b, a, c) = Pf(a, b, c), j(a, b, c)-f(b, a, c)

= N f(a, b,

c),

where P = 1+(ab), N = l..;... (ab) and (ab) denotes the operation of interchanging a with b in the function j(a, b, c). For n letters such a P has n ! positive terms, and forms the positive symmetric group, while N forms the negative symmetric group with the same terms, half of which have a negative sign. For n such elements a 1, a2 , ••• , an, Young writes P = {a1 a 2 ••• an}, N = {a1 a 2 ••• an}'·

In the above two-letter illustration it will at once be seen that, when/ is the determinant Ll, then Pl:,,,.= 0, NI:,,,.= 21:,,,.. Moreover, if any expression = ~ >ta 1 b2 c3 is taken which involves each letter and each suffix once in each term, besides a numerical coefficient .\, then the effect of the threeletter operators P and N upon are µ.Ll+ and µ.Ll, where µ. = ~A, and Ll+, I:,,,. are the permanent and the determinant respectively. These examples serve to explain how Young transferred the whole emphasis from the operand f to the substitutional operator, and very soon he had elaborated a considerable theory of these operators. The main result was contained in the method of the tableau (1900). A tableau is an arrangement of the variable in rows and columns, equal or diminishing both downwards and from left to right. A function, depending on, let us say, five permutable variables, possesses (among others) the tableau

xxii XX X XX

abc de .

From this model Young constructs the operators P 1 P 2 N 1 N 2 N 3 = {abc} {de} {ad}' {be}' {c}',

where each P refers to a particular row, and each N to a particular column. The sum of all 5 ! such expressions, due to the permutations of all five letters, Young calls T 3, 2 , the suffixes denoting the lengths of the rows of the tableau. Clearly there are as many such shapes (with the longest row and column always at the top and on the left) as there are partitions of n, the number of letters. For four letters there are five shapes XXX X

XX XX,

xx

XX'

xx X X

X X

X'

(I)

X

and therefore five operators T 4 , T 3,1 , T 2, 2 , T 2: 1• 1 , T 1, 1, 1, 1 • Young found that, for all values of n, a certain positive linear combination of these T's was identically equal to unity, say

l: Acp) Tep) = I, p

where each (p) denotes a different partition of n, and the coefficient Acp> is a non-zero perfect square rational number. In fact A

=

( II (ar-a,-r+s} 1/rr (ar+h-r}!) 2. ~•

r

where «r is the number of letters in the rth row of the corresponding tableau and h is the number of rows. This may well be called Y oung'a Theorem (1900). From it he deduced the Clebsch-Gordan series and a host of other results. It acted as a powerful crystallising influence by turning an amorphous function f, depending on several sets of variables, into the highly organized but limited varieties of forms Tf. Moreover, many such forms Tf vanish identically (as in the example above), and so do many products Tf then has exactly q2 arbitrary constants, where q is the number of standard forms of the partition (p ). As an illustration of this remarkable result, which may be called Young's Standard Theorem, we may take the above case of four letters. In the list of ten standard forms the sum of the squares of the subsets 12 +3 2 +2 2 +3 2 +1 2 must be 4 ! ; and in general ~q2 = n!. Also, in terms of the original coefficients Aep>, q = n! y(Aep>)The original Tep> is now replaced by a modified form

where each of r and s is summed from I to q, while ,\,., is numerical, and the P, u, N are definite substitutional expressions arising from the tableau. This leads to a matrix [A, 8 ] of q rows and columns which completely specifies the function T(pif(x, y, ... , z), derived from any function/ of n variables x, y, ... , z, which may undergo derangement. In fact the sum or product of two such modified T operators obeys the matrix law. For the general function /(x, ... ) the numbers ,\,.. in each matrix are quite arbitrary: so for four variables there are five matrices of orders 1 x I, 3 x 3, 2 X 2, 3 x 3 and 1 XI respectively, thus possessing altogether 24 elements. Furthermore, any modification of the function due to symmetry, or to skew symmetry, or to any other such property of the variables, is at once visible in the matrices-blank spaces appear. For example, if f(a, b, c, d) is symmetric in a, b, it is then capable of at most twelve values, by interchange of the variables in all possible ways. In this case Young found that half the rows of the matrices would be blank. Exactly which rows then sun·ive provided a very interesting problem, and it was analysed directly from the standard forms; for the case in point Young proved that this amounted to rejecting each standard form wherein the symmetrical letters a, b occur in the same column. A glance shows that this leaves only l, 2, l, l, 0 standard forms; and these tell us the numbers of non-zero rows, which have, of course, the same respective numbers of elements in them as before, namely, I, 3, 2, 3, I. Thence the full number of constants is found, by multiplying together respective pairs, to be 1+6+2+3+0 = 12, correctly. Young took a modest but wholehearted pleasure in these results, and his enthusiasm was infectious. "I am delighted", he wrote some months

xxvi

later (1926), "to find someone else really interested in the matter. The worst of modern mathematics is that it is now so extensive that one finds there is only about one person in the universe really interested in what you are". The tide turned in his favour with the appearance of the third memoir. Within a few years his method of the tableau and the standard theorem appeared in Weyl's Theory of Groups and Quantum Mechanics (2nd ed., 1930), as a means of elucidating the properties of quantum numbers, while the Clebsch-Gordan series, which had been largely responsible for substitutional analysis, was now found to be of fundamental importance for the whole of spectroscopy. Also during the last decade an interest in the theory of group characters has developed among several of the younger algebraists throughout the country. During the last fourteen years of his life Young wrote a steady series of papers elaborating his theory and applying it to the problems of invariants and their generating functions. In a letter to a friend, he wrote (1930): "For the last two years I have been working at a paper on the application of substitutional analysis to inrnriants, but though I have obtained a good many interesting results that I think might be worth publishing, yet I feel that it is too scrappy as yet to write out. . . . My ambition is at the moment to present the complete system for a single cubic in any number of variables. Quite do-able if things turn out as I hope; but I am a confirmed optimist, and so suffer many defeats". Something should perhaps be said in detail of this later work. The fourth memoir was mainly illustrative and technical, but it brought out the close relation between the matrices [,\.,] and the group matrices of Frobenius and Schur (1908). The fifth dealt with the group of rotations and reflexions of the hyperoctahedron in n dimensions. In the sixth a proof of Frobenius' generating function for characters of the symmetric group was obtained by the method of the tableaux, together with seminormal (or triangular) group matrices. The seventh and eighth memoirs, together with the communication to the Royal Society (1935), were devoted to invariant theory ; in the eighth was included an illustration from the invariants of a quaternary cubic, which has an irreducible system of six forms of degrees 8, 16, 24, 32, 40, 100. They had been discovered seventy years earlier by Clebsch and Salmon independently, but of this Young was unaware. The impression left by these examples, and by all the replies to his friends which he readily supplied on particular problems of geometry or invariants, was that in his hands the method of the tableau was irresistible. Known and unknown results alike were treated summarily and afresh; he merely reaffirmed (or corrected where they were wrong) the old and recorded the new.

xxvii

The motif which ran through these last few memoirs related to certain well-known forms called gradients (homogeneous and isobaric polynomials in the coefficients of the ground forms). Gradients were fundamental in the nineteenth century progress of invariant theory and in all the work of Elliott. Their behaviour at bottom depended on additive properties of sets of the positive integers occurring among their indices and suffixes. But so, also, were the properties of the forms which the method of the tableau produced. When the semi-normal matrices were used, Young found a marked parallelism in these two methods, which he called respectively the method of leading gradients and that of irreductible forms. Indeed, for the binary case, parallelism became coincidence ; and on this evidence, and his own instinct for algebraic truth, he surmised the following theorem: In general the complete set of leading gradients is defined in the same way as the complete set of irreductible forms. It is probable that his most recent and unfinished work deals with this, and in any case it is greatly to be hoped that the pages are complete enough to make their publication possible. Young's work is never easy . reading, for it lacks that quality which helps the reader to grasp the essential point at the right time. The very closest and constant attention is required to pick out some of the most fundamental results from a mass of detail. One could almost suppose that he camouflaged his principal theorems. His work resembles a noonday picture of a magnificent sunlit mountain scene rather than the same in high relief with all the light and shade of early morning or sunset. The craftsmanship is accurate and logical, and the ideas underlying many of the proofs are very beautiful. His powers of combining deep insight into abstract algebraic theory with an uncanny technical manipulative skill in all the practical applications give him a place unrivalled among his contemporaries. His humility, and perhaps his isolation and lack of teaching experience among undergraduates, prevented him from realising the importance of clarifying the crucial passage from abstract theory to detailed practice. It is hard to believe that processes which to the mind of Young were intuitively clear cannot yet be made part of our common mathematical heritage ; for if they can, then there is a great future for algebra. In drawing up this notice I have been very much indebted to l\Irs. Gunnery, l\Ir. G. H. A. Wilson, Prof. E. T. Whittaker, l\Ir. J. H. Grace, Dr. A. C. Aitken and l\Ir. W. L. Edge for supplying family, academic or mathematical details.]

COLLECTED PAPERS

The In·educible Concomitants of any Number of Binary QuartiC8. By A. YouNG. Received and read February 9th, 1899. The irreducible system is here arrive·d at by first finding the irreducible system of types and then the number of independent forms belonging to each type for a system of N quartics. Two concomitants are said to be of the same type when they can be obtained from the same form by polarization. For the purpose of discussing the system of types, a type is regarded as being of the first degree in the coefficients of each of the quantics concerned. The finiteness of the irreducible system of types has been established by Prof. Peano.• He proves that the complete system of concomitants for any number of binary n-ics may be obtained from the system for n n-ics by polarization alone; with the one possible exception of invariants @f the type

In other words, every type of a binary n-ic which furnishes no irreducible form for n n-ics is reducible, with the possible exception just mentioned. It was with the help of this proposition that some of the reductions for the quartic were first al'l·ivecl at; howe.er, other • A.tti di To1·i110, t. XVII., p. 580.

3

any Number of Bina1·y Quart,'.cs

1899.]

291

methods ha,e pro,ed shorter. The latter part of his paper is devoted to the discovery of the cubic types. From the fact that

A, As Bo Bi B2 B, Co cl c, c, Do DI D, D,

Ao Ai

is reducible, it is shown that all the types occur in the system for two cubics. His results areThe irreducible system for N cubics beiongs to 10 types, as follows:Number of Forms. Simplest Form. I__ Type. ___.I _______________ ,_______ _ I I

;

I

;

II. III.

.a, ,a,

One of the cubics

N

Jacobian of two cubics

(f)

Hessian of one cu hie

IV.

Third transvectant of two cubics

V.

Covariant order 3 of one cubic

VI.

Second transvectant of I. and III.

VII.

Discriminant of one cubic

VIII.

Jacobian of two forms III.

IX.

First transvectant of III. and VI.

X.

Resultant of two forms VI.

(Ntl)

(f) (Ni2) 2(Nil) (N;3) 3(Nt2) 4 (Nt3) (Nt4)

For the quartic, I have first expressed the types in symbols based on the quadratic. To do this, it is proved that the types of a binary mn-ic can be expressed in symbols based on the n-ic ; the symbolical factors being of the form of n-ictypes ; just as, in ordinary symbolical

4

292

Mr. A. Young on the Irreducible Concomitants of [Feb. 9.,

algebra, the concomitants of the m-ic arc expressed in symbols based on the linear form. It is easy then to show that there is only one type to be considered, of given degree and order. ,vriting this (abc ... k), the fundamental identities give relations of the form

(I+s.+s,+ ... +Sk)(abc

...

k)

=R,

where R stands for reducible terms, and S 11 S 2, ••• , Sk are certain substitntions. The chief advantage obtained from quadratic symbols lies in the possibility of using symbolical operators. with the help of which relations between forms of one degree and order may be obtained from relations between fo1·ms of the same order but of one degree lower. The invariant type of highest degree I 6 has been expressed in terms of determinants of five rows and columns; by means of this a number of syzygies may be at once written down, in fact I6 P equals a sum of products of forms, there being at least three forms in each product, where P is an irreducible form of any type except 12 and I 3• 1. Consider any simultaneous system of binary nni-ics,

... , ...,

x B ,,.,..)(_Xi,

X2

)"'"= b"z"•'

where and the identities are taken to define the relations between the symbolical letters a 0, a 1, ••• , and the actual coefficients. Let f (A, B, ... , K) be a type belonging to this system; writing in this for A0, ••• their values in terms of the symbolical letters, f(A, B, ...• K) takes the form (a, b, ... , k), say. Now make any linear transformation, and denote by dashed letters the coefficients of the transformed q a.antics ; then f (A', B', ... , K') µf (A, B, ... , K),

=

where µ, is a power of the determinant of transformation ; hence also

(a', b', ... , k')

= p = ...

= k,

thi!; becomes l!' once more. 'l'he1·e is a fairly close C _ -

1

I lf

I

.. C•l}

{.,(ll.,(t/ w

a,J-1, ... K.

...,

... ••

{ l>(I)

'"

b, a, ... , n, b{ll, ... , /.,(•>).

Then, if Hno>,,, /,< 1', ••• , k1' 1 instead of the sets }' { a0 >... aH}

= (a-1) ! [1+ (a< >a< >) + ... + (aClla'•l)] {a/'l b''l ••• k< >}' {a11 >.. • a\•)} 1

= (a-1) !

2

1

a;=a

l {a(",lbl1> • •• k'-'l} {d'' ... a< 1>,br1>, .. . cqnirnlc11t to Ii. Since P does not contain 1J!.8• 1>, D 1' _ ( c1)1.(.a,1>) p ,,Cll1/"•ll - a, u '

1-ltc 1·ight-lmrnl i;iclc lwing 1111 lo11gm· a f'u1H'tio11 of

11< 11 •

Now, P is symmeh-ic in the sets a(I>, .. . , a; hence the functio11 Pis the same as (a"'h1.a• 1>) I', except that 1i< 1> a.ml nb(.B• 11)

C,JJl}! {l/%'tl .. or11+1J} (a(•lb (.c1,1;) }', which does not contain the i:;et 1/•>. In this the new set bf.B•IJ may be replaced by the old set 1/•1 by opemting with (a1"> v.a+ 1>), and the result becomes

where now a is to be rcga1·dcll as equivalent to b.

13. If

'l'., 0

= 8{a h }'{at hj' ... {a.b.}' {b ht ... h,,.} S 1

1

1

1} '

1

where

= S{a b {,ttbt}' .. . {a.b.}' {a,. . .a+1 ... a,.b S = {a 1n:1 .•. a,.} {b bt .. . b,,.},

then

'1~,o = .A 0, ,. 'l'0, 11 + A1, 11-i '1'1, 11 _ 1 + ...

arn] ;3>0, '1'.,.a

1 ...

b,,.} S,

1

+ A11 , u '1',., 0 ,

if m ,t:: n; b11t, if 1n.}' [f.3 + (a,,_ 11 ,1 h... 1) + ... + (a X

= 8 {a b 1

1 } ' • ••

therefore

.B ♦ 1 b,,.)

{a.b.. }' [-(1n-a) {a,._ 11 , 1 b ... i}' + (m+/3-a)] ...

a,,b 1 b 2

...

h,,.} S

(in-a) T.tI,/J-I + (m+/3-u) T..,.B-•; 'l'

_

•,.B -

m-a + T I 1n+fj-a+ l •,.B•I 1n+,B-a+I 7 ~+J,.B•

By repeated application of this formula, we obtain

Too=

1n +I 1

'l'o, 1 +

m

m+ l T1,o

= Ao,, 'l'u,;+ Ai, i-1 T1,i-1 + ... +A;, 0 T;,o, 72

J

{a,._ 11 • 2 ••• a,.b 1 b 2 ... h,,.} 8

x {a,,_,9, 2

=-

11 __

Mr. A. Y ouug on

128

[Nov. 8.

llXcept wl}en i>m, in which case the series ends witl1 ..-1,,,,i_,,.T.,,, i-m;

i being supposed to be not greater than n, and the A 'f., being numerical coefficients. To find these coefficients a recurrence formula is obtained by proceeding from the Inst line written down a step further. The coefficient of 7J,i-j+I in this will he

Aj,i-j ♦ I Hence

=

A., 11

1n

=

'm-f + 1 + 'I,.. ~·+ ':I ,)···•1j-1,i-j ♦ I+

?n

+.·t - 2;_J·+1

Aj,i-j•

m-n+ 1 1 +/3 -a + 2 A.- 1, 11 + ?n +a/J-o. A ... p-1•

1n

It follows from this that, if A

_ •• II -

(o.+/3) ,,n! (m,+ I +/3-:"). /3

(nt +J3 + 1) ! ' so long as /3 Def,= - l ( ,r, ;:;~ ·ui oy 1 O!Jz ~cf,= -

1 (

n

Let u 1,

tt 2 , u 3

3¢

Yi ~+!h V,l.' 1

~

and D

:i·u X:1, a:3 ,

) +.t·a v4> ~ , Vfh

34> 34> ) ~+!la~· v:i: 3

v.t 2

be three quantities defined hy

then and, just as in the fu1·mcr case,

'1',. • _,,F=n!ni!S{a 1 h1 }' ... {a,,h,,}' ( 'llb-b). m.l' 1ct 1 ... ct,, b1 ... b,. Jt ,: V y 11

x {a,,. 1 ••• a,. b,.. 1 ... b,,.} a,,. 1z ••• a,. z b1,,. 1V ... b.,.!I

= ci:u;•, and that we may write ci,. = a, b = b = ... = b,,. = b,

Let us now suppose that F 01

= a 2 = ... =

1

2

after all the substitutional ope1·ations have been performed on F; theu 2 2 1 11 1 '1',.,,._,, I!'= (n') 1n~ (m+n-·>h)' · (m!) (m-h)! - · (ab11)''a"'-"D"•- •a"" b"'v •,

au ha 1 ; «1 {:

and hence

and therefore, in order that both hand a 1 may be (r-1). If therefore n> (1·-1)2, every term is zero, and F itself is zero.

T,.b (b a,.• 1

1)

...

2

{u,.b,}' (b 1 11,.+ 1) 1'

{a,., 1 a 1 }' {u2 b2 }'

.. .

{n,.l,,.}' l'

= 0.

Hence

and thcrt!fot·e

NP. P

= ~2+ /:1. l NP { a h r-1

1 1 }'

= ...

P

= /3-J~!•t l NP {a, b

1 }'

{a:1b:1}' ... {a11 ,b11.}' P.

Lt!t {c,r:1 ... c11J be tlw group of next lowest orcle1· in P. 0

=

Nl' {11 1 b1 c1 }' {t1!b:1c:1}' ... {a,.b,.c,.}' {n,.+ 1 1,,., 1 }'

'l'heu

...

.. . {a11 , b11 , }' {ct,.. 1c1 c·2 ... c,.} P + NP {ct1 b1 c1 }' { a 2 b2 c,i}' ... {a,b,.c,.}' {a,.• 1 b,., 1}' •••

=

••• {a11J111.}' {b,.,1fi

··

{aH,b 11.}' P

= l3i-l3i+l # +l 2

... {

a11, b11.}' P .

(f3~-/3,+l)(/33-/3i+ 2 )NP{a b c }'{n hr}'

(/3a+l)(/3,+:!)

1

1 1

t t 2

• ••

.. . {aH,bH,cH,r p

_/32-/31+1 (/33-/3,+1)(13~-#,+2) /33-/3~+1 .81 + 1 (f3a+ l)(,8a+ ~) .8a-f:J1 + 1'

NI' {11 1/1 1'N

=NPNPN

= .A· 1 NPN.

Si milady,

'I'. P

=

.tt -,

P NP.

8. If P be expressed as a sum of 1mbstitntio11s, the coefficient of en.ch is unity. Similal'ly, if N he exp,·essed as a sum of substitutions, the coefficient of ea.di is ± l. Mo,·eovm·, if two ld.te1·s appear iu Uae sa.me cycle of one of the substitutions of J>, tlicy cannot appeat· iu the same cycle of a 1mbstitntion of N. Hence, if .~ be a substitution of P, s- 1 cannot appear in N unless s 1. The coefficient of the identical substitution in NP must then be 1. Again, the coefficient of every substitution in NP is ± l. Foe·, let t be any substitution of this product, and )pt it :t1'l'ive by taking the 1mbstitution i:1 of ."v with u 1 of P, so that

=

s,u,

= t.

Suppose that, it occnrs a second time in the sum, as, sn.y, s2 u 2• Then

= s~ut, ;;2 s u ui = 1. s1u 1

and the,·efo,·e

1 1

1

1

llut s; 1 s1 1s a fmhstitution of N, arnl u 1 u~-• 1s a substitution of P; hence

'l'he substitution t then can only occu,· once m NI'. rcma1·k applies to PN. Now

'fhe same

'L'=t{(11 1 at ··• a,.)}JNP:

the coefficient of the identical substitution iu 'I' must then be Hence we oht.ain the iclent.ity

.!. --

n.'

lA «1. a.!,

11 !.

... , t11,

by cquatiug the cocfficicuts of the identical substitution.

Remember-

99

Quantitrdive Sub., titutional Analysis

1902.J

369

iug that the opcratiom; are made in the order expressed by the equation we observe that, if S be any 1mhstitutional expt·csi;iuu, usu-•=

S

Hence, if

tu::J s.

= t{tt,a:1 ... ,t,.}::J S', 1rHu- 1

uS

or

=8

= Su.

It follow1,1 from this t,hat any one of the expressions T is commutative with any Hing-le substitution; and therefore with any substitutional expression w liatevei· whicl1 contain:,; no letters which do not appear in '/'. 11. Relati,,u.~ lwfween tliffi>reut 1!'11n1t.~ NP.

9. If

he any s11hst.it11tion, we ohtai11 hy mca11s of the '1' series

s

thus every i;ubstitntiou c1u1 be expressed in terms of forms NP.~ (01· PNs) . Between forms NP certain linear relations occur. To discuss them it is necessary to employ 11, notation which will completely define the fot·m nuder diseusi;ion. We will \\Till'

=PN. Here the 1·owH iu·c takc11 to define the positive symmetric groups, while the colnmus define tlw negative symmetric groups. Again, we will ,Vl'it.l•

. .J,

111 J Ill ,, .. ... .... al

l. . . . ..

ltt , lll:.!,:.!· · · '12,,. ,

fl

100

I,. I · · ·

a

·/1,

a1,

... , ) '

••{

-J

= ,a, . 1 ,,1, ~ ··• "1, tt,,'', l

= 1,,rJ->,

}'{ «1

11 t., 1 • •• 11·2, «;!

" ' tt1, o }' { 11 1 l

•

A

,

(It

•

1 ... £t1,

,

}' •••

i} "

0

{

lit ..

~

, .... ,

370

(Feb. 18,

Mr. A. Young o'II,

where t.he rows define t.lie negative symmetric groups, the columns the positive symmetric groups. The notation use ,·)

a2, I •·· • ·• ••• · · ·

a,., 1

u-1

zero.

:l :

•..••. ••• n,., /J,·

l{

This follows at once from § 4.

Similarly, the product (s > r)

a::~..·.· al,

... . .• .. •. ..

ah, I

r •• • lt1,

/J1

"a~ ,· ...

lt1

,

aa,. , '1'

J

aa

r ,.,

lt>. .}'

= 0.

'

cr

= P (b b

1 2)

u,

and consider the following cases.

(i.) Let. tl1en

1r

(li 1 li 2 ) u u

then

( h1 b2 )

= (al,

1 b2c ... ) ;

= (ctb c ... ) = u", 2

= (ab c . .. dh 1

u

u'

= I.

2 1'f ... ) ;

= (ab ej ... )(b c ..• d). 2

1

Let the greatest g1·onp of I' in which any of the letters of u appear be {a 1 a2 ••• 1i,.}. Suppose that of these letters au a 2, ••• , a1, appear in .~, necessarily all iu different cycles. The cycle in which ah appears we will call cr1, and let Now cr1 may he written in the form

u; ((t1, b),

where u; is 1t cycle which contains the letter b, but not the letter anrl where b helongs to a group of P whose degree is :I> r.

102

a,,,

[Feb. 18,

Mr. A. Young on

872

Then

PsPN= Pu'u; (a,,b) PN

= r+l-h l Pu'u;[(a,,b)+(a,..

1

b)+ .. . +(11,.1')] l'N

= r+ : -h Pu'u; [-(I+(a b)+(a b)+ ... +(a 1

2

1,. 1

b))

+{l + (a 1 l,) + ('s'uN.~.

17. To illust,rate the ln.i;t few parng-mphs, we will consider one or two special forms P N.

Let then

{:t J 2

PN={~:a2

(a 2 a3)

5;

= a. }{a a~}' (a a = {a a,}(a a ){a a = -{a a }(a a ){a a {111 2

1

1

he11ce

Agnin

108

1 2 }'

2 8

1

Again,

2 8)

2

1

3

1 2 }'

S78

Mr. A. Young on

[Feb. 13,

hence

=fl~.,.

Again hence

that is

t {U1U:1}' [1 + (a~a3) + (a~u.,) + ... + (a~an-1) r-1 s111 n.3n, ... a,._I Sl = 0. ( llta,.

In general it is to be observed that, if a, b be two letters such that the groups of Nin which they appear are each of degree ..I''N',,, where {a 1 ,, 2 ••• a,} is a factoe of every P', and, moreover, is a factor of tlmt gmnp of I'' which is of cleg1·ee /3, /3 being the degree of the g1·eatest. g'l'Oll)l of P which contains one of the letters a 11 a2, ••• , ct,.. Fo1-, if a 1 is the lett«!r which appears in the g1·oup of P degree /3, then {a 1 a 2 ..• a,.} PN

= [l + (11,,1t 1) + (a,,11 2) + ... + (11,.11,,.1) l [l + (n,,.1 p

where (HG)~= pHG-a relation c cxp1·essed in tei·ms of snch ns have a 1 a11d a,._ 1 zel'o. Hence, if n < 4,, tlie1·e is not 11101·11 t.l1an mie invariant for c:wh deg,·ee.

28. Although the aho,·e identities contain all tl1e linear relations hctween tl1e inv:u·iants f, yet there n.1·e of hm· relations--not independent of thcsc-wl1ich it is m;efnl t.o have. Consider the resnlt of multiplying

f ,, 111.1 ... a,,, 1 = 1 ( /,I

b~ ... b,,,

s

I

by any positi\·e symmeti·ic g1·oup r. ·lve may regai·1l tl1e pm1l11d ns the sum of the results of operating on J with each of t.he snhslitntions of r; or else we may multiply the ,rnhst.itutiorntl p:wt of J by r as cxplaine1l (:-\ection III.), and so exp1·es:-1 the product as a i,;11111 of such terms as I, hut each of which Imm all ilie lette!'s of l' in the Ka.me row; the two expressions fot· the pr01l11ct mnst he e1pml. In particnlar, if the deg1·ee of I' ix greater than ,n, then the secornl exp1·ession is zero. Co11:-iitle1· now the inval'iant

f (a,1,

c:i 1,

... ,

a,.)

= HG],

11,Jl(l Jet ui; find the resn It. of wt·iti11g I' I fo1· I. Let the degree of l' he -w, and let ns sn ppose that the lcttcr·s of r ocen py p places i II the uppe1· 1·ow of 1, and -w-p in the lowet· row. 'l'hc11 l'/ is tire s11111 of the results of atTa11gi11g tire -w letters of r in nil possible ways in the

w places assignc" +... -n,. 2

In this case the transvectant (I.), and every one of its terms, belongs to the second class. When n 1, n 2, 1is are all greater than \ 2 +µ, ancl A2 +µ is not lass than 23 - 2 , we may express the covariant (a1 aJ"' (a2 a:i}"' linearly in terms of the covariants*

... , '-+ ", (a 8a 1)...

(a1aJ"2 +"',

(a 8a 1)'-+ ... ,,._ I (a1 aJ,

... ,

(a 8a1 )"• +,,.__•'· 3+ 1 (a 1 a)·••·S - l :

(a 1aJ"2+,,.-i (a2aa),

... ,

(a1aJ 2'· 2 (a2a 8)"•+,,.-2'· 2•

Now, hy § 3, if C be any one of the covariants in the first two rows just written down, the number of factors invoh-ing a1, a2, a8 only in the transvectant (C, a:;-"'q~)~!,,, can be increased. • Stroh, Math . .d.1111., Bd.

142

XXXJ . ,

pp. 444-454; Jordan, Lio11ville'1 Jo11r. de Math., 1876, 1879.

1908.]

207

COVARIANT TYPES OF BINARY n-lCS

Hence these covariants can ultimately he expressed in terms of covariants of the second and third classes, and of covariants which contain the factor (a1 aJ·\ where X1 s ; then, as before, if we put a [a 1 a 2 ••• a,_ 1], we have to consider the form

=

(aa."(•- 1 (ua., 11{ '

•••

(aa.)'"·-•.

By Stroh's series this can be expressed in terms of 1. (a.a.,+ 1t,+"'.•- 1[-.,_ ,- 1(ua,+ 1) .... , ... , (ua,)"·' (aaH 1t,+>-,- 1-"·' (aa.+ 2t"'

....

We proceed to i;how that each form in the fil'Ht two rows is reducible. (1) Any member of the first row is

([a,as+1], [a.a,+1J'>", - R say, = (a,as+1 )"'.,-1+>-.,-v = a,as+1 = aas+2)"'••I ... (aaK)"•-I ,

[ a.as ➔-1 ] -

where and

[

]t -

/J,

(

so that we need only consider the form (a/3)" (aaH~t .. 1

•••

(aaJ• .. ,.

In addition to the actual reductions implied by hypothesis, we must also consider thoim rednctionH which co11HiHt i11 i11creai;i11g the earlier indiceH n.t the expense of the later ones. ('fhiH ii; 1t common feature of all inductive proofa, since these reduced indices m1ty be raised to their proper limits by virtue of the inductive proof ; D. Note, p. 225.) So, since we have assumed the theorem true for the index ;\, (of a factor followed by not more than (K-s-1) other factors), we must have J I ~ a-,, s+I, if the form last written down is irreducible. Hence every form in the first row is reducible. (2) If we put

tRas+2)>-.. 11Ra )>-.. 2 ° • \}J tRa )>-•-•.> [a] , -= \}J \f~ S+3 '$

K

0

and we have to consider forms

tRa tRa$+2)>-.,., VJ ...)" \I-'

...

(Ra)>-.-, fJ It.

•

'£he condition for the irreducibility of this form is by hypothesis JI~

i.e., 150

JI~

sum of µ's whose first suffix is .~, and which do not contain it suffix s+ 1, a-,-a-s, s+I•

1904.]

227

P~mPETU.\NT SYZYGIES

Hence every form in the second row is reducible: and, since the index of («a,) in the last row is in every term ~ ... The syzygy {a 1a2 a3 a 4 f .. 0 will reduce the form (a 1a 4)2 (a2as)"'-2, 1 i;ince (a1a 4)(a2 a 3)"'~ C1 Ca, hy the Jacobian identity; the syzygy ja 1 a 2 a 4 aa} .. - 0 reduces i11 the same way the form (.. ~ 2,

µ ~ 2;

.>.. ~ 8,

µ ~ 2;

;\~4,

µ~ 2;

and so the generating function for all products C~ is

0

~

~

0-0

- - + - -2+ - -2 = - -3 (1-x)2 (1-x) (1-x) (1-x)

•

Finally, when the weight is unity we have only forms Oi 0 2 ; by the Jacobian trani;formation, we can compel any one definite letter to appear in the factor C2 ; so for weight unity there are only three such forms, while for all other weights there are six: the generating function for the products C~ C2 is therefore

ux2 /(1-x)+Sx. 153

l\In. A. YouNG uml Mu. 1'. W. Woou

280

[May 1:l,

The results for degree 4 may be summarized thns :Form~.

Ut•nera.tiug FunctionH. ,1'. 7

(1-x)a

4x3

(1-.x)2

6x2

1-x

+ax

1

Hence the genemting function for the total number of form:; of degree -1 ii;

III.

DEFINITION OF R1muc1BILITY.-GENBIL\L l\h:Tnoos oF Rt:vuc'fION llY DIFFERENTIAL OPER.\'f0RS.

5. The product forms will be arranged in a definite i;e(ptence to be particularized immediately, and a product will be defined 11,i; reducible if it is linearly expressible in terms of product fonns which follow it i'n this definite sequence. The sequence of the letters involved in a i;ymbolical expression will be that defined by their suffixes, viz., a 1, a2, • • • , a 8• The criteria for determining the relative positious of any two products A and B in the sequence will be made to depend successively on : (i.) The partial degrees of the factors of each product. (ii.) The arrangement of the letters among the factors of each product. (iii.) The weights of the factors of each product. (iv.) The indices of the symbolical determinants of the factors of each product. (i.) If

B 154

= C,.l c... .. . c,.,,

190-l.]

281

PERPBTUANT SYZYGIES

then A 1>recedes B, if the lirst of the quantities

which is not zero, is positive. A_

(ii.) If

B

o,,,,o,,., ... c,,,_ l

= c:,,, c:,., ... c:... ) '

so that the factors of A a.nd B are of the same p1trtial degrees, then the arrangement of the letters is taken into considemtion.

C,,, 1 contains the letters a,., ar,, ... , ar,,,,,

Suppose

c:,,,

and where r 1, r2 , such that

••• ,

r,,,,, or

and

contains the letters (t.,,, (t.,,, 11 1, .~ 2 , ••• ,

< s1
I : {P2 (cb)"};

(I.)

and this syzygy is a reln.tion hetween productH of covn.riant:; of the form 1\ 1\ and the same product:-; having the letterH b and cl interchn.ngetl. Now we It.we arrauged our products iu a fixed sequence, we shall SU})pose that the prodnctH on the right-haml side of the relation (I.) come after those on the left in this fixed sequence. 'fhe syzygy may then be conveniently written where R denotes any products which come in the fixed sequence after the products on the left-hn.nd side : the sum of the product:; on the lefthand side is, in fact , according to our definition, reducible. 7. In the first place we shall consider the reduction of products C2 C.: let C2 C. (ab)" (d. dl'' (cl. d 2)" (cl. d._ 1 1 : the se11uence of :;nch product:; is primarily determined l,y the letter:-; contn.iued in the factor C2 : we suppoHe tl1at the product in which tl1e letters are ab precedes any of the productH in which the letters of C2 n.re

=

0

ad 1, ad2 ,

. ..

adr

.. . ,

t·-

respectively.

'rhis is in accordance with our defined sequence of product:; when all the letters d 1, d2 , • •• , d, come after the letter b. 1n this case we have a series of relations of the form e11 , C.

where

= R,

= R, .. . ' a ZJ D,. = ~d - X 2 ~ l • e1'' (d.d2t" .. . (d.d. - l·-',

hut it may Le eW,+E,+ .. . ➔ r:,>c,

= 0,

= Jl.

Now C. may be represented as a sum of trnu:;vectants of coniriants of a definite set of r of its letters with covariants of the remainder, and so we may also obtain syzygies of the form c.

c~,2

(a1 a2)

>-c·•;.

Now any of these proclncts is reducible if it is expressible in term:-; of products following it : thus for (a1 a 2)A C~ there is no such rednction, so that;\~ 2, and the generating function for such products is 1x 2 V 2 , where -x V2 is the generating function of C~ ; for (a 1 a 3)A C; there is a reduction corresponding to the interchange of a2 and a 3 , so that ;\ ~ 8, and so on ; finally, for (a1 aJ>-c; there are reductions corresponding to the interchanges of a 6 with a 2 , a 3 , a 4 , a 5 respectively, and therefore A ~ 6; hence the generating function for all these products is

x2 +x3 +x4 +x5 +x6

_

x2 -x7

_

- - - - - - - - ' - - V2 - - - - 2 V:J 1-x (1-x)

Assume that the generating function for V

_

,n-1 -

Then the products

c~•

( a1

X

2,,,_ 2

x 6 (1-x)(l-x3)(1-x5) o (1-x)

c;•- • is

(1-x)(l-x3) ••• (l-x 21n- 3) (1-;i:)2m-2

·

are

a2,,,-1 )

Acm-1 2

,

Here, as before, any product is reducible if it is expressible in terms of 1 there are (r-2) reductions products following it : thus for (a1 art corresponding to the interchanges of a, with a2 , as, ... , a,_ 1 respectively, and so A~ r; for, by Section III., these reducticms are all independent ; the

c;-

168

1904.]

245

PBRPETUANT SYZYGIES

generating function for all products

xr

•

1s -1 V,.,_ 1 , and therefore the -x

generating function for thiA Aet

c;' is

_ x 2m(l-x)(l-x8) ... (1-x 2"'- 1) (1-x)2m

'l'herefore, by induction, the generating function for products

c;• i1-1

x2m(l-x)(l-x9) ... (l-x2,,,-1) (l-x)2,,,

(2)

Generating Function for c;'P8 _ 2m (where P8 _ 2 ,., is any form or product, other than C3 or C~, wkich contains neither C1 nor C2).

16. Lemma.-If the roots of y"-Pi!l"- 1 +p2 y•'- 2 are 1, x, x2, ... , x"- 1, then

_ W (l-x")(l-x''- 1) ... (l-x"_.,+1)

(1) (2)

p., - x

(1-x)(l-x2)

•••

(1-x.,)

•••

•••

=0

·

p.,, +(1-x) Pn-2+(1-x) (1-x8) p,._, + ... +(1-x)(l-x3)

+(-)"p,.

(l-x"- 1)

= 1,

·n even;

p,.+(1-x)p,._ 2 +(1-x)(l-x8)p,._~+ .. . +(1-x)(l-xa) ... (1-x"- 2)Pi

= 1, ·n odd.

(1) Assuming the theorem true for all values up to and including n, we shall show that it is also true for the value (n+ 1) ; it is obviously true when n 1, 2.

=

If

y"+l_q1y"+q2y"-1+ ... +(-)"-lq,.+1

= (y"-P1Y"- +P2Y"- + · ·· +(-)" p,. ) is

its minimum weight is

(2"-2), the generating function

-(2•-r•)-(2•-r•)-... -(2•-r·-•) ,c-1 ,r-2 1 ,

take all values satisfying

< rl
05.]

81

CERTAIN (:LASSES OF SYZYGrns

'fhe generating functions for products c;• urny be calculated m the s1u11e way, e.g., thnt for (a 1aa)A (a.2 aJ" (a 4 as>• is

x 9 ( 1-x"•- 3)(1- x"•- 5)(1- x".- 4) (1-X)3

for this product is always reducible if ;\ "~n.-2.

~

n 1, if µ

~

n 2 - l,

01·

if

20. A covariant of degree 3 can be expressed in terms of members of the second class if its weight is greater than the order of any one of the quantics concerned. The generating function of comriants C8, the a;", 11 1 l> 11 2 l> n8 , 1s !Juantics concerned being a~•, a'.!', -z It

;r

x 8 -(11 1 -2)x"•+(n 1 -8)x" 1 + 1 (1-x)

Consider products C2 C3 , in particular the set (a1 a~• (a 8 a 4 )" (a 3 as)",

where 11 1, n 2 , syxygies

11 3,

n 4,

n5 are in ascending order of magnitude.

The six

reduce all formi3

=

where rr 1, 2, 3, -1, 5, 6. The argument for the linear independence of these syzygies is identical with that for the independence of perpetuant syzygies. We thus see that all products (a 1 a 2t (a 3 a 4)" (a 3 a 5)"

are reducible for which X+µ > n 3 -7. The products when the letters are arranged in any one of the other possible manners may be treated in the same way. In § 12 (p. 240) of the paper on " Perpetuant Syzygies " a table is given of the limits of the indices ,,, X, µ for irreducibility in the various cases. The general result here is that, if the product is reducible when " ,C:::: 111, then it is also reducible when X+µ > n-111, where n is the order of the quantic of lowest order that occurs in 0 3• To prove this fact, it is merely necessary to remark that the syzygies 209

82

CERTAIN CLASSES OF SYZYGIES

which give the reductions in the first case are the same (except for their weight) as those which give the reductions in the latter case. 21. The main theorem of this paper applied to a single quantic shews that its covariants may be treated as perpetuants so far as the known results for perpetuants as regards reducibility or syzygie& are concerned, provided that forms having a factor (ab)!" are neglected. The method by which this result has been arrived at is of such a general nature that it would appear almost certain that when the, as yet unknown, syzygies of ..,. = 2&-r-l

+ Er + E,·+1 + ··· + f&-1

, ... .. .. . .. .. .. ...... ... .. .. (iii)

X. = 26- 3 + E, + E, + ... + E6-,

x, = 26-• + 2 (E. + E. + ... + E&-,) + E, where the fs are positive integers or zeros!. It has been pointed out that this result, which was proved originally for perpetuants belonging to a single quantic (in which case E, must be even), also gives the conditions for perpetuant types when these are expressed in terms of product.'! of either of the forms (i) and (ii), the sequence of the letters not being fixed§ . • In writing down symbolical products we aball omit factors of tbe form a 1z, a..,, ... . t Grace, Proc. Lo11.d. Math. Soc. , Vol. xnv., p. 107.

::: Ibid. p. 319. § Grace and Young, Algebra of I nvariants, p. 379.

211

l\lR YOUNG, ON RELATIONS AMONG PERPETUANTS The exact number of perpetuant types of degree

( w-

2

1- 1

+ 1 + o-

0-2

o and

2)

67

weight w is known to be

·

But when the sequence of the letters is not fixed it will be found that the conditions (iii) give too many perpetuant types ; it is the first object of this paper to determine what , are the relations among these forms. It will be convenient to make use of the notation of the theory of substitutions; to avoid confusion symbols which refer to substitutions are printed in Roman type. The symbol {ab ... kj i11 used to denote the sum of the substitutions of the symmetric group of the letters a, b, ... Ir. The symbol {ab ... kj' denotes the sum of the substitutions of the alternating group of the letters a, b, .. . k minus the sum of the substitutions of these letters which do not belong to the alternating group.

In the lo.st part of the po.per the rnducibility of certain transvectants is deduced from the results obtained. I.

Consider a perpetuant of degree

o

P =(a,a/' 1 (a.a,)"-' .. . (a,_1 ad•a-1. All perpetuants such as P will be supposed arranged in order according to the indices of the different factors, the sequence of the letters not being fixed. Thus if

Q =(b1 b.}"1 (b,b.t• ... (b,_1 b,}"a-, where b,, b,, ..• ba are the letters ai, I½, ... a, arranged in some order; then Q will precede P provided that the first of the differences µ,, - >-11 µ,. - X., ••. JJ,a-1 -

x,_,

which does not vanish is positive. If all these differences are zero, P and Q belong to the same set, and take the same position in our arrangement. To express that the sum of certain forms like P is equal to a linear function of perpetnnnts which precede them in the chosen arrangement and of products of forms of lower degree, it will be convenient to write

"'i.P=R. The symbol R is throughout used in this sense. conditions (iii) we have

Thus unless the indices X. satisfy the

P=R.

2.

Covariants of degree three.

All perpetuants of degree three can be expressed in terms of those of the form (a 1a 2)~ (a,a 3)", X { 2µ,.

212

68

MR YOUNG, ON RELATIONS AM.ONG PERl'ETUANTS

If ;>,. = 2µ., If X=2,u+l,

(a1 a 2)'" (a 2 a.3 )" - (a,a3)'" (a,.a,)" = R. (a,a.)"'+ 1 (a.,a3}" + (a.,a3)""+1 (a,a,)"

These fact.~ are well known.

+ (a3 a 1)'"+' (a,a.,)" = R.

They may be deduced at once from Stroh's series•.

The relations may be written (a,a.)"" (a.a..)"'=! {a,a.,a,} (a,a2)"" (a.a,)"'+ R,

{a.a.a.,)' (a,a.)'"+' (a.a.)"= R. When X > 2,u + 1, we have three independent forms (a,a.)A (a 2a,)". Hence the number of perpetuants of degree three and weight w = 3k + 2 is 3k ; for we may take µ. = 1, 2, ... k.

The number of perpetuants of weight w = 3k + 1 iii 3 (k - 1) + 2 = 3k - 1. number when w=3k is 3(k-1)+1=3k-2.

And the

In every case the number is w - 2; and this is known otherwise to be the exact number of perpetuants of degree three and weight w. Hence there -can be no relations between these perpetuants other than those just enumerated. 3.

P =(a 1a/'1 (a.a,l• ... (a&-,ai·s-,

Let

o,

be a perpetuant of degree whose indices X satisfy the conditions (iii) : we 1>roceed to prove that if E,-, = 0 (r > 2), then

la.a.+,}' P = R.

Let the symbol a refer to the perpetuant (a 1 a,>°''1 (a.a,l• ... (a,_,a,_,f-•

when considered as a single binary form of infinite order.

Then

(aa,/'•-• (a,a,+,l'' ... (~_,a6 }"s-, - P

Now >.,_1 = 2a-,-, + X,, since

E,-, =

= R.

0, hence by Stroh's series

{- I (x,_,i+ x,) (2x,;\,.- i) (aa,)ll,_, +ll,- i ( a,ar+t )i =O

x (a,+1a,+ 2)ll,+1 (a,+2a,+./•....-, .. . (as-,as/a-,

But

= 0.

. ......... .. ..... .. .... .(iv)

(a,a,+iY'' (a,+1 111)"2 (a,+,a,+l,..., ... (a6_ 1 a6is- 1

= (a,a,+,Y'' (a,+,a'f• (aa,+ /r+• ... (a 6_,a,/a-, + R; 2

and when /J.t < 2.i-r-, the perpetuant on the right-hand side can be expressed in terms • Math. Ann., Bd. 31; Algtbra of Invariant,, p. 64.

213

69

l\lR YOUNG, ON RELATIONS AMONG PERPETUANTS

of forms which contain a greater number of factors involving a,. a,+i, a only. Thus we see that all the terms of the second sum in (iv) may be included in the symbol R; m fact this relation becomes

Whence {a,a,+i}' P

and therefore

+.

When

E, =

= R.

0, wc have A., = 2;\.,.

In § 2 we saw that where a, fJ are any two of the letters a,, a., a3 • Hence also (a,a.)2'11, (a3 a,/' (a.,a,}"• .. . (a6_,a6}"6-,

for

Therefore when

-

(a,a.,)2'11, (a,,ai' (a,a,)~• ... (a 6_,a6}"6-•

= R.

E, = 0

{afJ}' P

= R,

where a, fJ are any two of ai, a2 , a3 •

Silnilarly from the fact that we deduce that if

{a1a,a,1}' (a1a.,)1A,+I (a,a.)•• = R,

E, =I

{a,a,a.)' P = R.

The following relations have been obtained: (a) (b)

(c)

(d)

Er-•= 0, E, = 0, f. = 1, E, even,

{a,a,+1 1' P = R. {a,a.,)' P = R, {a2 a.)' P = R. [a,a,a.,}' P = R. {a,a.,)' P = R, E, odd, {a,a,j P = R. (r > 2),

It remains to shew that there arc no more relations. 5. Assuming that the relations just enumerated are u.11 that exist between the forms which satisfy the conditions (iii), we proceed to prove that the number of these 2a-i + . depen d ent 1s . . . ht an d ",orms wh'1ch are J'mear1y m _ 1 + 0 - 2) ; w bemg t he we1g

(w -

o the

O 2

degree.

Let The conditions (iii) become µ, { µ, 1= µ, .. . -I=

µ6-J,

µ, { 2µ,

together with the fact that the µ's are positive integers.

214

70

MR YOUNG, ON RELATIONS AMONG l'It:RPKTUANT:;

Consider first those fot'ms fol' which JJ,t+i

= 0 =JJ,r+o =•••=JJ,8-i,

and µ.i, µ,.,, ••• µ., arc nil different from zero.

P

Let

= (a,a,>''• (a,a,)>..' •. • (as_, ad's-,

be one of these forms ; then by § 3 {ab}' P=R,

when a, b are 11ny two of the letters a,.+2, a,·+s• ..• a,. The letters

a,, ct,, . . . a,+

1

can be chosen in

(r ! 1)

ways. When this set of lettel's

has been selected, the number of forms corresponding to given values of µ.,. µ,.,, ••• µ., depends (i) on what consecutive pairs of µ.'s are equal, and (ii) on whether µ,, - 2µ,. = 0, 1 or > 1. This number is then quite independent of o, provided that o;:;: r + 1. Also the set of values which can be given to (µ.,. µ,., ••• µ.,) is independent of

o.

Hence if tf, (1·, c:r) is the number of independent forms P of degree o fo1· which µ., > 0, and P.•·+i = fl,,•+o = ... = P.s-, = 0, then tf, (r, c:r) is the number of independent forms of degree r + I, for which no µ, 1s zero, and Iµ. = c:r. Iµ.

= c:r,

Now if µ,, is not zero, set of values (µ,,, µ,., , ..• µ.,) to the set of val11ei1 (µ., forms of degl'ee 1· + I, for

the numbel' of fot'ms of degl'ee 1· + I, cort'esponding to a given of the µ.'s, is the same as the numbe1· of fol'ms col'responding 2, µ,., - 1, ... µ.,. - 1). Hence tf, (1·, c:r) is the total number of which 'i.µ. =c:r - r - I.

Again, the total number of forms of degree o is equal to the numbe1· of forms for which µ,, is the last non-zero µ., together with the number of forms for which µ. 2 is the last non-zero µ., and so on. Hence this number

Now we shall assume that the total number of forms of degree weight w is + + r - = (c:r' + r (w r-2 r-2

2,-, I

2)

Then by hypothesis

when r

+ I < o, and when r + I = o, but

'liT,

+s-2)

0-2

'liT 1

o

and

2) ,

where c:r' = Iµ.: also that the number of forms of degree (

,.
,. = 7,

{a,a.,] C = 0, .{a,,a,] C = 0, (1 + (a,a..)(a,,a,)] C = 0. Also C can be expressed in terms of forms (ab)• (be)• ( cd),

and of products of forms.

C= .A,T,C+ A,,,1'.,,,C+ IA {o.bl'{cdl' (ac) {bd) C +R.

Hence

Rut from t,he above equations

T,C=0,

1'.,,,C=O ;

and we have only to consider expressions like

!n,a.,J' [a:1a.)' {a,a,) {a,a,) 0= ½la,a.,]' {n,o.,}' (a,a,] (n.,a,) [1 + (a,a,)(a,a,)] C= 0. Aod hence C is reducible in this case. (iii) Transvectants

C = ((a,a.),

(a3 a,)')A.

If X = 1, 2, 3, C is reducible owing to the fact that the weight is less than seven. If ;>,. = 4, and n, = n, , C is reducible since

\a,a.] C = O. (iv)

Transvectants

C = ((a,a,'f,

(a,a,)•)A,

If ;>,. = 1, 2, C is reducible owing to the fact that the weight is less th ..n seven, If ;>,. = 3, and

n, = n, = n 3 = n,,

(v)

Transvectants

(vi)

Transvectants

C is reducible since

[l + (a.,a..) (a,o.,)] C = 0.

C is reducible if ;>,. = 1, 2, 3, 4, aod n, = n.. n3 = n,. C = ((a,a.)•,

(a,a,)')A.

0 is reducible if X = 1, 2 and n, = n,. (vii)

Transvectants

C = ((a,a,)',

C is reducible if ;>,. = 1, 2, 3, and n, = n, = n, = n,.

218

(a,a,)1 )A.

ON BINARY FOR)IS

By A. YouNo [Read January 22nd, 1914.l

THE object of this paper is to develop a method of attacking some of the problems in the theory of binary forms. Problems connected with the enumeration of complete systems are particularly in view. Every method introduced requires some justification for its existence ; its utility needs to be judged by results. In this case the method is at once applied to covariant types of degree four of the hinary form of order n, and the complete irreducible set of these is obtained. The preliminary analysis is concerned with the theory of perpetuunts, and incidentally the complete system of perpetuant syzygies for every degree and weight is obtained. It appears that all perpetuant syzygies of the first kind can be obtained symbolically from those due to Stroh. and that consequently the extension to any degree of the work"" of )Ir. Wood and myself, for the first eight degrees, depends solely on accurate enumeration, and does not require the introduction of any new principle or the discovery of a different type of syzygy.

I. Explanation of M etlwcl. 1. We are concerned here entirely with the symbolical notation. Its introduction by Aronhold at once gave a methocl by which all covariants could be mathematically expressed. At the same time in the calculus it provides every form considered has the covariant property. But it has the drawback that a great many unnecessary forms appear in any discussion. Yarious methods have or can be suggested by which the forms considered may be limited to a linearly independent set. But such methods cannot avail much in most problems unless it is possible to express the product of two forms so expressed in terms of the corresponding forms. • Proc. London Mnth. Soc., Ser. 2, Vol. 2.

219

442

[Jan. i2,

Dn. A. Yomw

Grace,* in applying the symmetrical notation to Mac!\Iahon's theory of perpetuants, has succeeded in doing this for the case when the order of every quantic considered is infinite. In this case he selected one quantic a~ for particular attention, introducing the symbol a 1 into every determinant factor, by means of the equation (a2a3)

ll1 6

=

(al a3)

a2z -(al ll2) ll3,.

Thus the only symbolical products he had to conside1· were of the form (omitting factors a.r) (a1

at>"

2

(a1 a 3)"3

• ••

(a 1 a,)"•.

These, when perpetuant types are under consideration, are all linearly independent. There are no superfluous forms. Now, when we come to forms of finite order, we cannot, as a rule, apply this method as it stands, for the reason that there are not a sufficient number of factors a 11 in order to be able to introduce the letter a 1 into every determinan~ factor. In fact, if we c11,n do so, 11 1, the order of the conesponding quantic, must be equal to or greater than the weight of the covariant considered. Let w be the weight of the covariant C, then if we multiply C symbolicall Jv by a.,_,.., we can express aw-ni C in the form ~ ~

where N is numerical. ,re have thus, as in the case of perpetuants, a linearly independent set of symbolical products to consider. But there is this difference : separate products do not represent actual covariants, but only certain linear functions of such products. We shall proceed to shew how every such product may be made to represent a covariant or else a form which we shall call a fundamental form. After that we shall proceed to shew how prnducts of covariants may be dealt with, as in the case of perpetuants. 2. Let us consider covariant types of degree

o;

• Proc. London Math. Soc., Yo!. xxxv, p. 107.

220

that is, co,·ariants

191-:l.]

ON BINARY FORllS

443

linear in the coefficients of each of the quantics

It is supposed, to start with, that these quantics are arrauged in a fixed sequence. Let us fix our attention on some covariant type expressed in the ordinary mannet· as a single symbolical product. We say that this covariant is a term of the continued transvectant

(using the single symbolical letter to denote the corresponding quantic). This statement is nearly obvious. An immediate proof is obtained by induction. Assume it true for degfee o; then, if C be a symbolical product representing a covariant of, degree o+ 1, C is a term of a transvectant and, since P is a symbolical product representing a covariant of degree &, the theorem in question is true for P, and therefore it is also true for C. Now the fact that every term of il. transvectant differs from the whole transvectant, by a linear function of transvectants of lower index, leads us at once to the fact that any term of th, continued trans,·ectant (( ••• ((a 1 a 2)>- 2, a 8)•\

\

'ai••, ... , a6)~,

differs from the whole transvectant by a linear function of forms

which are such that the first of the differences

which does not vanish is positive. We are then at liberty to express every co,·ariant type of degt·ee in terms of continued transvectants of the aborn form.

o

3. Let us now retnro to the consideration of a single symbolical product which represents a covariant type C of degree o. Let the weight of C be 1c. The symbolical product at-" C can be expressed in the form I

221

444

Dn. A. YouNo

[.fan. 22,

where N is numerical: by repeated use of the equation (ara,) a1.

= (a.a,) a,, -(a1a.-) a_,z•

We shall arrange the products in a definite sequence by saying that

precedes provided that the first of the differences

which does not vanish is positive. The continued transvectants will be supposed arranged in sequence according to the same law. Now it is to be observed that a continued transvectant is defined by the same set of numbers ;\2, ;\3, ••• , ;\1 , as a product (a1 a 2)"2 (a 1 a3)Aa ••• (a 1 aa? 1•

If the continued transvectant be expressed as a sum of the products considered (by multiplying it by a.;0 -n1), the first of the products in our • sequence to appear will be that which is defined by the same numbers. Now every continued transvectant represents a covariant type; but only certain linear functions of the products (viz., such as are divisible by -n represent actual covariants. The difference between the two cases

a;

0

I

1)

being accounted for by the fact that there are certain limitations to be imposed on the indices of the transvectant ; whilst the only limitations to the indices of the product are those expressed by the inequalities ;\ 2

)> n2 ,

;\ 3

)> n 3 ,

. •• , ;\ 8

)> na.

These limitations are also necessary for the transvectant, but m addition we must have (i)

;\2 )> n1, 2>- 2 +;\ 3 )> n 1 +n 2, 2>- 2 +2;\a+>- 4 )> n 1 +n 2 +11 3 ,

••• ,

2\i+2;\3 + ... +2Aa-1+>-a )> n1+n2+n 3+ .. . +na-1• In the case of products we shall use the term fundamental forms to denote products for which the set of inequalities (i) is not satisfied. -!. We proceed to shew that corresponding to every other product, that is to every product for which the inequalities (i) are satisfied, there is

222

445

1914.]

a unique covariant which can he represented as a linear function of that product and of fundamental forms. We have seen that the trans\·ectant

can be expressed as a linear function of our products of which the first term is Let be the next term in the order of our sequence to appear; if it 1s not a fundamental form we may subtract the coval'iant N(( ... ((a 1 a,i)''i, a 3)"3, at)"•, ... , aa)"·

from both sides of our equation. Proceeding thus step b)· step, we arm·e at the truth of the abo\·e statement. That the covariant is unique is eddent from the fact that every covariant can be expressed in terms of the trans\·ectants considered, and that these trans\·ectants can be expressed in terms of the co\·ariants found, and i•ice vetsa. 5. Let us use the notation

to denote the co\'ariant corresponding to

i.e. the covariant obtained from this product by the addition of a linear function of fundamental forms. Then we have a set of linearly independent cornriant types of degree o in terms of which e\·ery such covariant type ma~· be linearly expressed. And this set is composed of the forms (\1, Xa, ... , ~),

where and the X's further satisfy conditions (i). It will be convenient to hM·e a notation for the covariant (:\2, Xa, ... , Xa),

in which the letters corl'esponding to the different quantics appear ; we

223

[.fan. 22,

Dn. A. YouNo

4413

shall for this purpose use the notation (

aA,aA• 2 S

•••

a1

a"~ 8

= (;\2, A3, •.. , X,).

-

In order to discover what forms are reducible, or to find relations between products of forms, it is necessa1·y to he able to express the product of any two of our forms as a linear function of the forms of a higher degree. Thus, for example, the product 2 3 • •. aA') 8 (a"•a"• ~

)" _

(aa+1aa+2

-

') ( a"•a"•3 • '• .,)

••• ,

= R,

representing the fact that this form has the quantic The next equation will be

e-a,D, (0, >.3, or (0, X3,

••• ,

Xs)-X,(1, X3,

••• ,

••• ,

Xi)

a; for a factor. ~

= R,

Xs-1)+ (;0)(2, X3,

... ,

>.,-2)- .. .

= R.

This equation with the help of that already used reduces (1, >. 3, ••• , Xs-1) ; i.e., it expresses this form in terms of eal'lier forms in the sequence and of prodncts of forms. We next consider and it is easy to see that this reduces the form (2, >. 9,

••• ,

>-,-1-2, X0).

When we come to our next equation

it is necessary to take it in conjunction with the last. tracting, -1,,D,_ 1-P,D1_e-a,iD1 _1] [e -

+>-s

(0

'I.

'I.

We ha,·e, on sub-

'I. )

, A3, • ••• A6-h /\4

(;t)-s-1-2, >. 0 -l)- ...

+terms in which the last argument is less than >. 6 -1

=R. Also [e- 0 :v,- 1 -l](0, X3 ,

••• ,

>-s-1+1, :Xs-1)

= -(Xs-1+ 1)(1, Xa, ... , >-s-1t :Xs-1) +

)(3, X

-(>-s-~+ 1

= R. 226

3 , ... ,

(>-,-~+ 1)(2, X3, ... , >-,-1-l, Xi-1)

>-,-1-2, As-1)+ ...

1914.]

ON

449

BIN.\UY FORllS

Using the results of our first two equations "e may wdte these two equations Xa-1(2, X3 ,

••• ,

X,-1-l, Xa-l)-e·;- 1) (3, :\3 ,

••• ,

Xs-1-2, Xa-1)

= R,

1-2, X,-1)

= R.

(Xs-~+ 1)(2, X3•••• , Xa-1-l, X,-1) -(Aa-~+ 1)(3, :\3,

••• , :\ 6 _

These two equations are proved to be independent by calculating the determinant formed by the coefficients-its value is ½Xs-i e•s-~+ 1). Thus we can express (2, X3,

••• ,

Xs-1t :\s)

and

(3, X3,

••• ,

Xa-1t Xs)

in terms of forms and of products of forms ; where µ 2 -.(a 1a 5)A• ••• (a 1aa)"•,

234

(a3a4)} >-. (a1 a/·• ... (a1 a,,>-,,

= {(a111 8)+(a2a4)}>-fA,, {(a 1a3,)-(a 1ai!)}"•, ...

{ (a 1 a,.,)-(a 1 a 2) }"•. (a 1 a,)"·• . ..•

Then we have a set of syzygies we will call syzygies of the type D : such are { (a.2a,,)-(a2aa)} A,, (a2 a,)A,,

... (a 2a,/•• (a 1a, 1)",·. (a 1a,.)"··, ...

= (a a,/'•• { (a1a,.)-(a1aJ }A,, •.• { (a1a,.)-(a 1a2)} "•• (a 1a,.i>"•·,(a a,)''-·· ... . 1

3

\Ye obtain fresh syzygies by replacing any term (a 2 a,) on the left by {(a 2 as)-(a 2 a 3) f, and making the corresponding change on the right of {(a 1a,)-{a 1a 2)f into (a 3 a,). Or we may change on the left (a 1a,.) into {(a1 a,.)-(a 2 a 3)}, and at the same time on the right (a 1 a,) into { (a 3 ar)+ (a 1 a2)}. In this way we obtain a set of syzygies which will give us the 25 - 3 relations between our equations e-a,D•• -P3 D•:.i -

... -a:,D

•J.

[O , O, A,1, '\

'\

• • •, 1\8

J = R,

where rr1 , rr2, • • • , rr,, are any, all or none of the numbers 4, 5, .. . , Again, we have the syzygies of the type C,

o.

{ (a 1 a,.1)+ (a2 a 3) J"··, { (a.1 a,)+ (ct 2 a 3) JA,, (a2a,)"•, ... (aia,/'• (a 1 a,)",, .. .

= {(a2a, )+ (a1 as)} A,, { (a2a,

2)

1

for example.

+ (a1 as)} A,., (a 2a, 2)A,, ••• (a 2 a, )"•• (a 1a,)",, ... ,

•

This will give the relation '\ e+a D,. +o,D,. [O O /\4., 3

242

I

~

f

f

'\

J -_

• • • J 1\8

R•

ON

1914.]

BINARY

465

Fom1s

And so we obtain syzygies which yield '\ ] - R

ea,D., +a3D., +... +,,.v.,, [0 0 '\ '

' 1\4, ••• , " '

-

'

where 0"1, 0"2, ••• , O"t are all or any of the numbers r 1, r 2 , r 3, •••• We have then certain syzygies which we shall include in the type D; an example of these is { (a1 a.,)+(a2 aa) f>-,, { (a 2a.,)-(a2aa) f "•·((½a,")",_ ...

(a 2 a,/•• (a 1 a,2)",, (a 1 a,)"•·, ...

This particular syzygy yields '\ ea3D,-a3D, [0 0 t /\4, I

'

f

R•

• ••t ' /\'5] -

The syzygies of which this is an example yield the set of relations , ea3D,,+a 3 D..,+ ... +a,D,1 -a-JD0 ,-a3D0 , - . .. -a,D01 (0 , 0 , /\4,

where P1t p2 , S1, Sg, ... ,

s,,.

••• ,

Pk are any of

r2 , r8 ,

1\,

••• ,

and

... ,

'•] _ R' /\0 -

0"1, 0"11 , ••• , O"k

are any of

Lastly, we have a set of syzygies we shall call syzygies of the type E. They are really forms of the Jacobian syzygy, an example of these is ((½aa) j (a2a,.}-((½aa) }",. { (a2a.J-(a2a 9)}"•,(a 2a,y,,

...

(a 2 a,/'-• (a 1 a,)",, (a 1 a,2)",, -

•••

(a1as) j (ai a,2) - (a1as) f ",, { (ai a,.)-(a1as)}"•· (a 1a,l·•·, (a 1a,.)",·, ...

(a 2 a,.)"•• ... (a 2 a,.)". -(a1 a 2)(a3 a,,)"., (a3 a,,)"•• { (a 1 a,.)-(a 1 aJ f "••

...

{ (a1 a,.)-(a 1 aJ f "•• (a 1 llr1)"•·, (a 1 a,)"·,, ..• ,

e-o•D,.-a,D,~

whence

(0 , 1t '\/\4,

'\ ] -_ R •

• • ., 1\6

and so, in general, we have syzygies which yield -a,D00 -a3 D0 , - ... -a.3 D0

e

where

0"1, 0"2•••• , O"k

2&-t+I

,,.-113D0 ,-113D0 , - ... -a,D,1,

V

[O ' ] , I\,, ,

, ] -_ R '

••• ' I\&

are any, all or none of s2, s9,

18. We have to prove that the (i)

•

i

••• ,

s.,.

equations

(0 t 0 t ,1\4,

'L] -- 0 t

• • •t "I

243

466

DR. A. YouNo

where

o-1, o-2 , ••• , o-k

[Jan. 22,

are all, any or none of t+l, t+2, ... ,

o;

(ii) where pi, p 2 , ••• , Pk are all or any of the numbers r 1, r2 , ra, . . . which are contained in t+l, t+2, ... , o; and o- 1, o-2 , ••• , a-,, are all, any or none of the numbers s1, s2, ••• , s,, which are contained in t+ 1, t+2, ... ,

o;

where o-1, o-2 , ••• , o-k are all, any or none of the numbers .~1 , s2, which are contained in t+ 1, t+2, ... , o:

••• ,

s,,

are just sufficient to express all forms

[0, X3, X4,

••• ,

Xi, X1+1, .. . , X6],

for which X3 < 26- 1+1, in terms of forms

where µ 3

,. and is not one of S1, S2, .• • , s~, then this form has been reduced by a previous equation. But, in either of these cases, there is a syzygy by means of which this equation can be expressed in terms of previous equations, as we have shewn in our theorem of § 10. Thus, to every equation we have a definite reduction or a syzygy. 21. Now let us review the perpetuant types of degree

o.

Firstly, they can all, reducible or irreducible, be expressed linearly in terms of the forms and these forms are all linearly independent. Secondly, any product of perpetuant types of total degree o can be expressed as a product of two perpetuants, neither of which is necessarily irreducible ; and, when this product is expressed in terms of our standard forms of degree o, it can be written, without ambiguity,

e - a,.D,'-a,.D,s- ... -a '.D,

246

'7

(' .., A2, .. . , Ar-1,

O, Ar+l, ..,

• ••t

). -)

''-a

0:-:

1914.]

469

JIINARY l'OR:\tS

Thirdly, the complete discussion of the equations e-ci,D,,-11,.D.,-... -n ,,D,,

(A~, ... , Ar-1, 0, Ar+h ... ,Ai)= R,

involves, firstly, the discovery of the laws of reducibility and irreducibility, and, secondly, the discovery of all the syzygies of the first kind. The laws of reducibility established by Grace follow from this. And we have now shewn that all syzygies of the first kind can very simply be deduced from those of Stroh and the Jacobian form of syzygy.

III. Forms of Fin ite Order. 22. The discussion for forms of finite order follows identically the same lines as that for perpetuants. We express all covariants of degree o in terms of the forms

defined as in § 5. \Ye then consider every possible product of two covariants of total degree o, and we express it in terms of our standard forms. The equations which we get in this way will give us the laws of reducibility of our standard forms, and also will yield e,·ery syzygy for this degree. The discussion is rendered more complicated by the fact that (A~, Aa, . .•, Ag)

is no longer equal to the simple product (a 1 aJ~' (a 1 aa},\ 3

• • •

(a1 aal"',

but is equal to this plus a linear function of the fundamental forms. If the set of inequalities A2 )> n1, 2X 2 +X8 )> n1+n 2, 2A 2 +2A 3 +X 4 )>

11 1 +11 2 +11 8 ,

•• • ,

2X 2 +2Aa+2A4+ .. . +2Aa-1+Ao )> n1 +112+n8 + . .. +na-1t

is not satisfied, i e itself a fundamental form ; and we must write (:\2 ,

Aa, ... , Ag)

= 0.

The analysis for perpetuants must then be modified in two ways. 247

470

Dn. A.

Firstly, the product (s

[Jan. 22,

YoGNG

< s1< s2< ... n 1• We proceed to prove the following theorem : When the orders n 2, 113, • •• , na of the quantics concerned are g.reater than the weight of the covariants under consideration, while the orde1· 11 1 is less than this quantity, the covariant (A 2, As, •.. , .:\6) 1nay be represented by the sum (a1 aJi..2 (a 1 aa)"• . .. (a1 aa)",

where

For simplicity we will take the symbolical identity

o = 4.

And for this case we will prove

(I) (a 1 a,J"• (a 1 aa)"' (a1 a 4t•

+r(-)' (ni-~z+li-1) (~a) ( A~ ·) (a1a2)''•+; (a1aa)i(a1a4)p-i-i iJ p-'t-J

where The forms on the right are ordinary symbolical products which represent as they stand covariants of the quantics with which we are concerned. Let us assume the truth of this identity as it stands and then deduce that it is true when A4 is changed into A4 1 and n 1 into

+

249

472

DR.

A.

[Jan. 22,

YOUNG

ni + 1. It is to be noticed that this change leaves p unchanged. To do this multiply the supposed identity by (a 1 a 4). Then when the order of the a1 quantic is n 1 + 1, those terms under the sign of summation on the left for which i 1 are no longer fundamental, and thosP. terms only. From the identity

=

(azaaY

(a2a4)P-i

= {(a1aa)-(a1a2) fi {(a1a4)-(a1a2)f p-i,

we obtain (a1 aJ (a1 a4)P-i

= (a2aaY (a2a4)P-i_

l:

(-)i1+i2

i1+i2'1'0

(~J (p-:-j) i1

(ai a,Ji,+ i2 (a1 as)i-i, (a1 a4)p-j-i,.

i2

We make use of this result and the identity becomes

+ ~

i 1 +i2 =#:0

=

"iP /=0

+ t~1

(~a) ( _ A4_ ·) (-)i1 +i2

J

p

1 J

(!) (p-:-j\ (a a 2t1+l+i1+i2 1,1

)

1,2

1

X (a 1aJ- '• (a 1 a 4)P-i-i,

(p-p-1 +1 f) (a2aa)P (a1 a2l••+t (a1as)Ao-p-• (a1 a4).... +1

()s) (p~~ :_) (a2a3)i (a2a4)P-i (a1 a~"2+-'s+f-p (a104).._.+1-f

The right-hand side of our identity is already the same that we should get by writing X4 1 for X4, and n 1 1 for n 1 in the identity we want to prove. The coefficient of (a1a 2) " 1 +1+• (a 1al (ai a,)l•-i-i on the left is

+

+

+ (_), l: (Xk k

250

3)

(

X 4

)

p-1-k

(

k) (p-i-j p- k ) .

j

1914.]

478

ON BINARY FOR:11S

f (~s) (p-~ -k) G) (/

Now

4

i

k)

= the coefficient of ziyP- 1zP- i -i in the expansion of

= the coefficient of ziy,- 1zp-i-j in the expansion of 1+ .3.JL} >-, (l+y)>-•+>-• (l+z) 1(1+ _EL_}>-,~ 1+y I 1+y Hence the coefficient of

+

The identity is true, then, when we replace \ 4 and n1 by \ 4 1 and n1+l. If, then, it is true for certain values of \ 4 and ni, it is still true if these values are both increased by unity, and therefore if they are both increased by any the same number. (i) Let 1'1, be greatel' than \ 4• Then, if the identity is true when 111 - \4 and O are written for n1 and \ 4 , it is true as it stands. It will be sufficient simply to discuss the case \ 4 0 and leave n 1 unaltered. The identity then becomes

=

(11)

(a1 a,i'2 (a 1 a-i)"•

+!(-}' (n1-~2-+;_i-l)

(P Xa 1)..a-p)( P )]. '1 p-1 ,,-1 p-1 11-.>..a+P

To find the value of this we shall prove the identity

- (p) ( . p ) + (p + 1) (. p ) (~) i 1 t-1 2 i-2

.•• + (- )i (p +1-1) (. p .) J t-J

Assume that it is true as it stands and add

one

more

term

(~+j) (. ~ )

to each side. 1+1 t-J-1 The right-hand side becomes

(-)j+ 1

(-).i+l

(p+J)! {pi-(i+lHp+j+l-i) t = (-)i+l (i-1) (p+j+l) (j+l)! (p+j+l-i)! (i-j-1)! i j+l i '

and Ro the induction proceeds step by step : for the identity is obvious for j 0. Making use of thia result we find that the coefficient of (a1 a2)",+11 (a1 as)".-11 is

=

which is the same as the coefficient of the corresponding term on the lefthand side of the identity, for .>. 2 +.>.. 3 n1 +p. This coefficient is unity 1, 2, .. . , n 1 -.>..2 , and its value is when ,, is zero, it is zero for ,,

=

=

for

The identity

(I) is then true if .X4 = 0,

and therefot·e whenever n 1 >

(ii) Let n 1 be equal to or less th_a n 252

.>. 4•

.>. 1•

Then, if the identity is true

ON

1914.]

BINARY FORllS

475

when O and X4 -n 1 a.re written for 111 and X4• it is true as it stands. It will be sufficient to discuss the case n1 0. It is just as easy to take the case n1 < \ 2• Here the left-hand side of (I} becomes

=

every other term under the sign or summation vanishes. The left-hand side is therefore zero. On the right there are no terms in the first sum, for \ 3 -p is negative, and in the second sum every coefficient is zero fo1· f-1 < p-1-j, since j must be less than \ 3• Thus (I) is true when 11 1 < X2• [In the same way we see that the general expression in the enunciation of our theorem

vanishes when

11 1


\ 4 ; it is therefore true for all values of 11 1 and \ 4• Now the identity (I) expresses the sum of

and certain fundamental forms as a sum of symbolical products which represent actual covariants of the quantics under discussion. This sum of covariants is then the covariant we have named

The theorem is then true for degree 4. Assuming that it has been proved for degree -1, it can be proved for degree 8 in just the same way that it has been pro,·ed for degree 4. The actual form of the covariants on the right 1 of the identity is not given, and it is not required. It is sufficient that the right-hand side of the identity should contain only symbolical products which represent actual covariants of the quantics concerned. There is no difficulty in obtaining the expression, but it is troublesome to write out, and no advantage is gained by doing so. 24. When the orders of all the quantics are finite the case is not so simple. For the discussion of the covariants of degree 4 we require 253

476

Dn. A. YouNo

[Jan. 22,

the linear function of fundamental forms that must be added to (a1 a~A2 (a1 a3}A3

a;•;, a;;, a;:.

in order that the sum may really be a covariant of prove that:-

We shall

The covariant (X2, Xs)

= (a 1a~>-, (a1as)>.a +!(-)' (Aa-P1-:--P21+i-1) (Aa-P_2) (a 1a 2?•-P•+i(a1as)P,+Pt-i, i-

where p;

P1-i

= :\ +X -n;, or 0, according a.~ X2+:\3 > or < ni. 2

9

In the first place the terms under the sign of summation are all fundamental forms, for 2 (11-i-p2+i)+p1+p2- i

=

2n1+p1-p2 +i

=

11 1 +,½+i

> 111+11 2,

since the coefficient is zero unless i > 0. Moreover the index of (a1 a2) never exceeds n1 -p2 +p 1 n2, for i :t> Pi• From the identity (II) of the last para.graph, we have for the case

=

P2

= 0,

+i-1) (PlXa

(a a )>-2 (a a )A•+!(-)• (Xs- ~1 1 2 1 3 i-1

.) (a a )"• +i (a a

-1,

1 2

\P,-•

1 SI

-1 + c-) = t~o (plpi-1 (a2Cl3)P1 (a1 a2)A2+f (al a3)A•-P1-t, A3-p,

6"

an identity which establishes our theorem in this case. We shall take this as it stands and suppose that n2 has its least possible value :\2 +X3• Now in this replace X3 by X9 -p2 , keeping X2 and p 1 unaltered; then n 1 must be replaced by n 1-p2, since n 1 X2+X9 -p 1, and n2 must be replaced by n2 -p2 ; we have

=

l) (Xa-P.2) (a a )n,+ i-P2(a a )P•-• ( 1 + c) (a a )P• (a a )>.,+t (a a )>-.-p,-P,-t =>.s-P,-Pt ! Pi-

(a a \>.2 (a a )Aa-,2+!(-); (Xa-P1-:--P2+il 21 1 s i-1 Pi_ 1

t=O

S"

Pl - i

2 a

l

1

2

2

l

3

1 -a

Now multiply this result through by (a1a-a)P• and we have (a1 a~>-• («1 a-a l•+I(-)• (Xs-P1-:-Pl2+i- l) (Xa-p:) (a1 aJn, +i-p, (a1as>P,+p1-i i-

254

P1-i

•

1914.J

477

ON BINARY FOR~IS

Since the t·ight-hand side of this represents a co,·ariant of the quantics Q. E. D. concerned, it is 2, \J.

= (:\

25. It will be sometimes useful to use the notation

(a.a~"• (a1 as>""+!(-); (Aa-P1-:-Pl2+it-

l) (As-P_2) (a1a2t•+i-P2 (a1aJP1+,.,-i p.-i = (a aJ,., (a as),.,. 1

1

Also we shall copy the index notation of ordinary algebra further by writing {(a1aa)-(a1a~h:

= !(-)i (~) n+n 1, we have, from (YII),

(Ag, 0, A4) (for ;\4 cannot exceed

u4 = n in any case).

Thus always and

(X.2, 0, A4)

= R,

1, A4)

= R,

(;\ 2 ,

A4 n+11 1 -1.

80. Let us now discuss the reducibility limits of ,\~. We have the following equations (YI)

(YIU)

when X3 )> 11 1, (IX)

(0, A3 , A4)-X3 (1, A3 -l, A4)+ (~)(2, X3 -2, ;\ 1) - ( ~3) (8, X3 -8, A1)

+ ... =R .

)(3, A -8, ;\. )+ •••

(X) (0, >-3 , X1)-X3 (1, A3 -l, A,)+(~) (2, A3 -2, >. 1) - ( ~3

-A4 (1, ;\.3, A4 -l)+A3 A1 (2, X3 - l, X4

3

1

-1)-(~a) X (8,A -2,A -l)+ ... = R. 4

3

1

when 2;\.3 +A, ► 2n. Taking these last two equation!,! together, we see that (IX) is tt·.ue when either ;\4 )> 11 1, or 2X3 +X, )> 211. And that when we replace the~e conditions by the original condition of (IX) we may replace (X) by (XI) (1, A3 , A4 -1)-,\3 (2, A3 -l, A,-1)+ (~3)(8, ;\3 -2, A1 -1)- . . .

when X, )>

11 1,

= R,

and 2X3 +X 4 )> 2n.

259

482

DR. A. YouNo

[Jan. 22,

Let us first see what the equations give us just as they stand. (0, >.3, >.,) is reducible if any one of our equations exists. Hence we see that it is reducible unless ).3 > n 1 , >., > n 1, and 2A3 +:\, > 211. The reduction of (3, A3, A4) requires the coexistence of equations of each of the four types, and there is only one way in which it can be reduced. It is easy to see that it is not reducible unless ).3

►

n.-8, ).3 ► n 1 -8, A4 ► n-1, >., ► n 1 -l, 2A3 +A, ► 2n-5, 2A3 +>., ► n+n1-5.

The conditions of reducibility are more complicated when X.2 it will be convenient to separate the discussion into two cases.

= 1 or 2 ;

(a) n 1 ~ n.-The equations (VIII) and (IX) always exist; together they reduce (1, X3 - l, >.,). Then (1, As, A4) is reducible if A3 < n. From (YI) and (VIII) we have a reduction for (1, As, A,), proYided A4 < n and 2As+A, ► n+n1 -l. Thus (1, A3, A,)= R, when As< n, or when

X,


.3, X,) is always reducible.

= R,

(/3) n 1 < n.-Here (1, X3, XJ may be reduced by (YI) and tYIII), in which case we have the conditions (i)

Xs ► n11

>., ► n-1,

2A 3 +X, ► n+n 1 -l;

or by (VIII) and (IX) in which case the conditions are (ii) or

(iii)

X3 ► 1ii-l,

X, ► n 1 ;

X3 ►

2>.3 +X, ► 2n-2;

11 1 -1,

or else by (XI) when (iv)

X, ► n 1 -l,

2X3 +>., ► 211-l.

Also (2, X3, X,) may be reduced by (VI), (VIII) and (IX) wi1en the conditions are (i)

X8 ► n 1 -2,

>., ► n-1, 2X3 +X, ► n+11 1 -3;

or, by (XI), (VIII) and (IX), when (ii)

260

Xs ► n 1 -2,

X, ► n 1 -l,

2A3 +X, ► 2n-3;

0~

1914.]

nINARY FOR)IS

488

or else using (VI) and (YIII) to reduce the first term of (XI), we obtain the conditions

81. It is necessary to examine equation (VI) a little more closely. The two conditions fo1· its existence may be replaced by the single condition

>-s ::I> 1'1_-p.

When >.3

= n -p, 1

the equation takes the form

and when >.3 < n1 - p, it takes the form

where in each case R represents a linear function of products of forms and of forms (0, µ 3, µ,) for which µ 4 < >. 4 -1. A difficulty apparently arises when we use (VI) and (VIII) in conj unction in the case >. 3 n 1 -p ; for eliminating (0, >.3 , >.,), we have

=

giving a reduction for (0, >. 3 + 1, >. 4 -1) instead of for (1, >. 3, >.,-1). But in this case (0, >.3 +1, >. 4 -1) is reduced by another equation of the type (YIII), unless p 0, and the reduction of (1, >.3, >. 4-1) then follows.

=

and hence, from (IX), we ha,·e

Then, taking these equations in conjunction, we obtain the reductions exactly as stated in the last paragraph. 82. We haYe so far discussed our equations without any reference to the reductions already obtained when >.3 < 2 or >. 4 < 1. Thus some of our forms will be reduced twice over. In the case of perpetuants the result of equating the different reductions was shewn to lead to a syzygy in every case. :Now we shall find that it may lead to a syzygy or else it may lead to the reduction of a form not previously reduced. 261

48(

DR. A. YouNo Let us turn to equation (YI).

(XII) R

Put ::\3

[Jan. 22,

=0

and use (Y), thus

)co,

=

(0 , 0 , "4 '\ )+~( ( A4 . ~ - )i (A4-p1+i-l) • 1 iPi-l

111

+.i, Pi - i.)

+

giving a reduction for (0, n 1 1, ::\4 -n 1 -1) instead of a syzygy when ::\ 4 > n 1 1 ; it shoul n. Now this is already reduced by (IX) since 2(111 +1)+::\ 4 -11 1 -1 :t> 211. Also we have a reduction for the form (1, n 1, ::\ 4-n 1 -1) which occurs in this equation from (VI) and (VIII). Thus we obtain a reduction for (2, n 1-l, A4-n 1-l). This is the final reduction when ::\4 > 2n1, but if A4 :t> 2111 , we can use an equation of the type (XI), and so reduce the form (8, n 1 -2, A4 -n 1 -l). These forms were not reduced in§ 80. 1 is given by (VII). To find what (YI) The reduction when ::\3 gives us in this case, put ::\2 0 in (VII) ancl use (VI) fo1· each term, thus (assuming p 0)

+

= =

=

a"•1, a"• 2, (a 3 at• 4

= ~(-); (~t)[(a;a~•-;) a~•-}:(-Y(::\ -i~1> +j-l) (::\ -_i) a1

i

4

,

1

4

Pi-J

J-1

(0,

111

+j, Pi -j)]

since the coefficient of (0, n 1 +j, p 1 -J) is zero. Thus in this case we only get a syzygy of a very obvious nature. n, and then (VI) gi,·e3 When p is not zero, we haYe only the case ::\4 the reduction of (0, 1, n) which has not been reduced by (VII). When ::\4 0, (VI) only gives an obv~c::1s syzygy.

=

=

88. The equation (VIII) gives syzygies just as in the case of perpetuants when ::\ 4 0 01· 1, or ::\3 0. 1, we reduced the equation in the perpetuant theory by When ::\3

=

262

=

=

1914.]

485

ON BINARY FORlJS

means of the syzygy

This holds good as it stands when A4 ::t> n-1, and A4 ::t> n1 -1. still furnishes an identity when A4 > n1 - l and A4 ::t> n-1. We write this identity

= (-)"• [ (::t !) {(a2a4)>---••• (a1 a 3)''•+ 1-

-(~: t !) {

But it

(a 3a 4)A,-n, (a 1a 2)n1 + 1 }

(a2a,)A•-"1-1 (a1 a3)"1 +2_ laH a.i)"•-111-1 (a1 a2)••1+2

J.

t + ...

Now, from (XII), we ha.ve (changing A4 into A4 +1)

:+l+1) {(0,

A ( 11

-(n 1

+1, A4-ll1)-(n1+1, 0, A4-n 1)}

111

+1)e~t!) {(O, n 1 +2, A4 -n1 -l)-(n 1 +2, 0, X4 -n 1 -l)}+ ... = R.

Hence on subtraction we obtain a syzygy if X4 ::t> n1 + 1 ; and a reduction for when The reduction equation is (XIII)

+ +}1) {[e-«•v•-1](0 , n +1

x_4 ( Jll

l

-(~:t!) l[e-a,D,_(n +l)](0,

•

X4 -11) I -[e-,IJ,_ 1](11 1

+ 1, 0, A -n 4

1)}

-[e-aiD•-(n.+l)](n.+2, 0, x,-n.-1)} + ...

= R.

1

11 1 +2,

>.,-n 1 -l)

"·ith the help of (IX), this in general will reduce the form (1, n 1

+1,

>. 4 -n1 -1)

when X4 > 2n1 ; but if otherwise we can use (XI) also and so reduce (2, ni, X4 -n1 - l). We must examine (XIII) further, owing to the presence of au excep263

DR. A. YouNo

486 tion.

[.Tan. 22,

Expanding, we obtain

(XIV)

_

;:i (-)' (~:tD[ (n ti;1

n1 A•-ut-j ~ i=l ;~o

I

(-)"+i

n

1)-1][(0, 1+i, (;.\

4

+ 1) I

\ 4 -11 1-i+l)

- n-1) we can use (X), and thus obtain a syzygy. This furnishes then no extra reduction when n 1 1. We have yet to conside1· the case \ 4 n, that is the equation e-a~D, (0, 1, n) R.

=

= =

84. The equation (IX) gives syzygies which are quite obvious when

\ 4

=0

or \ 3 < 2. For \ 3 2, we use the syzygy

=

{ (a1a,)-(a2as>}~+ 2 + {(a1a,)+(a2a3) t>.,+~

= l (a a 8

4)

+ (a1a,)f A.+ 2 + {(a 1a 8)+(a 2 a 4)}>-,+ 2,

which reduces the equation when

and \ 4 :t> n 1 -2. The equation exists only when \ 4 :t> n 1• We can shew then that thie furnishes a syzygy whenever our equation exists and \ 4 :t> 11-2. For \ 4

0

=

:t> n-2

1(a1a,)-(a2as> }"'+ 2 + {(a1 a.,)+(a 2aa) }"'+ 2

- {(a3a-1)+(a1 a~ }A.+2_ {(a2a 4)+(ai as)} >-,H

= P+2(a a,)>-.+ 1

(a1 a 11)>-•+ 2 -(a1as>>-•+ 2

2-

- (X,+2Ha1 ~>-.+i (aaa..)-(X,+2)(a 1aa)"'+ 2 (a2a 4)

(where P is used here and elsewhere to denote products of covariants)

= P+ {(a a )+ (a ct }}>-·•+s + l(a1a3)-(a2a,)rA n-2. We have still to consider the cases .\, n-1 or 11. In fact we ham to consider the four equations

=

(0, 2, n-1)

= R,

e-« 2 D3 (0, 2, n)

= R,

e-C1zD3

e-a,Do

(0, 3, 11- l)

e-"', 0 • (0, 3,

where it must be remembered in each case that

n)

= R,

= R,

.x, :t> ni-

85. The equation (X) gives obvious syzygies when ,\3 < 2 or ,\4 For the other cases the syzygies {(a 2 a 4)-(a,ia3)} 10

< 2.

= (aaa,)"', 265

488

Dn. A. YouNo

and {(a 2 a 4)-(a2a 3)} w-l (a 2 a 8)

= - (a a 4)'''3

1

[Jan. 22,

(a,1 a 2)-(a 8 a 4 )"' + (a 8 a.)'"- 1 (a 1a 4),

may be used, as for perpetuants; provided the weight w is not greater than 11. When the weight is greater than n we find ourselves with five equations to deal with of just the same type as those of equation (IX). We are thus left with ten equations to consider, four of weight n+l, four of weight n+2, and two of weight n+s. 86. For weight n+ 1 the equations, in the case of perpetuants were reduced by means of syzygies obtained from the symbolical identity (XV)

A 1(n+ 1) [(a2aa) { (a2a4)- (a2a3)}''-(a1a 3) {(a 1a 4)-(a1a 3)}"

+ («1 a2Haaal']

J

+A 2 [ { (a 2 a 4)-(a2 a3 )} >i+ 1 -(a3 a 4)"+ 1

+ A 3 [ { (a3a 4)-(a 1 a 2) f n+1_ { (a2 a 4) -(a 1 a 3)} "+1] + A 4 [ {(a3a4) +(a1a 2)} n+ 1 -

1(a 1 a 4)-(a2 a3 )} 11 +1]

+ .·1:; [ 1(a2a4) +(,,1 a:i) f n+l -

{ (a 1 « 4)

+(a2 aaJ }"+1]

+ (A2-As-A4) [ (asa4)n+I - { (a1 «4)- (a1a 3)} "+ 1] + n, we need the following results from § 25,

>

and

1(a1 a4) -(a1 a3)} n+ i _ --

11

-

•[(n +i-2) ( . 1

1

! ,+ (_ ) 11

!(-)'

i =l

1

L-

(1111)

(a1 a3)i (a1 a4};, +1-;-(a1 Ct4)11+1 -(a3a1)''-+ t

1) ] +nL. - 1) - ( _ ) (11+1-11 . 1 I111

111

Making use of these results in (XY), and of the corresponding resnlt for l (a1 a,)-(a1a~} n+1, we obtain from (XY) in the notation of this paper, (XVI)

-(A -A -A) 11

3

4

"-£'+ (->i[( 1

11

•=l

i~H-2) ( i-1

1) ] n_ )- 2n1 -1 ; but, if 11 ~ 2n1 -1, this is already reduced and otfr third reduction is (8, n 1 -2, n-11 1). When n1 n-1, we have to look for five reductions or syzygies; the three equations obtained for the general case enable us to express each · of (0, n, 1), (0, n-1, 2), and (n-1, 0, 2) as a sum of products. Substitute their values in (XVI) ; and it reduces to

=

-("! 1) {A +(-)''A ! c-a,l>,(0, 1, n) = P. 5

1

But using (VI) we find that c-a,D, (0, 1, 11)

= (0, 1, 1!)-(1, 0, 11) = (n-1)(0, n-1, 2)-(n-2)(0, 11, l)-(11-l)(n-l, 0, 2) +P =P.

And hence the extra equations give two syzygies here. When ,,,,_ n-2, we have one extra equation to obtain. It is plain that we must have A 5 - ( - )" A 1 0 in (XVI).

=

268

=

We find our

1914.]

Ois

4!11

BINARY FOR)IS

= =

= = (-)",

equation by putting A 1 A 3 0, Az A4 A~= 1. And by means of this we can obtain the reduction of the extra form (2, 11 -2, 1). 37. For weight n+2_, we fiucl that the equation

is requil'ed for the ordinary reductions, unless e-n2D3

(0, 2, n)

'!'he equation

11 1 ::;:;, 11.

=R

exists only when n1 ::;:;, n, and is then required fot· the reduction of (1. 1, 11). The equation

e-« 2v, (0, 3 , n- l)

=R

exists only when n 1 ::;:;, n-1; and

e-o,D,-a,P, (0, 3,

11- l)

= R'

which exists only ,vhen n ::;:;, 5 is reqnil'ed for ordinary reductions when ll1 < 4. Thus we require tht·ee reductions 01· syzygies when 11 1 ::;:;, 11, two when n1 n-1, one only when n-1 > 11 1 > 3, and none when 11 1 ~ 3. We replace n by n+ 1 in (XV) ; and observing that by § 25 we have

=

+ {n 2 - l - ( - ) " (2n+3) Ha 2 as)'1-1 (a 2 a 4) 3 - {n2 -n-3-(-)'' (3n+-t)} (a 2 aa)" (a:!ai.

and

= :•f (-)' (1itl) (a2a:ih+1 (a2a )3

+ {n-2-(-)

11

(811

+1)} (a2aa)" (a2a,)2. 269

492

[Jan. 22,

Dn. A. YouNo

In these we replace by (a2a3)"- 1(a 1a 4)3 j (a1a 3) - (a1 a 2)} n-I { 8(a1a 2)(a1a 4) 2 -8(a1a2)2 (a1 a 4)+(a1 a2) 3 f,

-

and

(a2 a 3)'' (a 2 (a2«a)" (a 1a 4)2-

by

aSJ.

j (a1aa)-(a1 a2)}" j 2(a1 a2Ha1 a4)-(a1 aJ 2 },

and then substitute in our new identity. In ordet that the identity may yield a relation between actual covariants, the constants must satisfy the conditions (XVII)

When n1 :::,,- n+2, we find if n is even no syzygies, but reductions for the fo1·ms (3, n-8, 2), (2, n-1, 1), (8, n-2, 1); and if n is odd there is a syzygy and the forms (3, n-8, 2), (2, n-1, 1) only are reducible. When n1 n+ 1, and n is even, our identity furnishes reductions for (3, n-3, 2), (1, n, J.) and (2, n-1, 1); but when n is odd there is a syzygy and reductions only for (3, n-3, 2) and (1, n, 1). When n 1 n, there are no syzygies, the reductions are (11 -1, 1, 2), (2, n-2, 2), (3, n-8, 2), when n is even, and (2, n-2, 2), (8, 11-8, 2), (1, n, 1) when n is odd. When n1 -n-1, we expect only two results from our identit~·, and we find that the constants must satisfy the additional condition A 4 +As 0. And whethe1· n is odd or even we find the new reductions to be (8, n-4, 8) and (1, n-1, 2). When n1 < n-1, we have one reduction only to look for, and we must have A 4 0 A 5 ; and therefore 2A 1 A. 2 2!1 3• We find then

=

= =

=

= =

= =

!

a reduction for (3, n1 -8, n-n1 +2>, when ·n 1 n1-l, 2:X3+:X 4 )> 2n-2, or ;\ 4 )> n 1 -1, 2:X3 +:X 4 )> 211-l.

...) :::,,. ,. 2n (lll n- 2 ::::,:, 11 1 > S,

2:X3 +:X 4 )> 2n-2 or ;\3 )> n1 - l , ;\ 4 )> or ;\ 4 )>

(iv)

11 1

= n-1,

11 1

= n-1 or n,

11 1 -1,

11 1

2:X3 +:X 4 )> 211- l.

a modification is introduced owing to the reducibility of (1, n-1, 2) ; we have then

2:X3 +:X 4 )> 2n-2 or :\3 )> n 1 - l , ;\ 4 )> 11 1 01· ;\ 4

(v)

11 1

= n+I,

(vi) n 1 > 11+1, (vii) n 1 > 2n,

)> n 1 -1, 2:X 3 + ;\4 )> 211.

;\3 )> n-1 or 2:X3 +:X 4 )> 2n+l. X8 )>n-1 or 2:X8 +:X 4 )> n+n 1 -l.

every form is reducible.

The reducibility limits of (2, :\3, ;\ 4) and (8, :\3, :\4) are traced in Figs. 2 and 8. It will be seen that in both these cases there is part

20

Flo. 2.

272

1914.]

ON

495

nJNARY FOR'.\18

of the figure which corresponds to forms irreducible for all values of n1•

tanh- (f/r) +(2( -'12> r/6 = and Again it is to be observed that Ba, A

=

Ba,~a

=

=

2a,

1 ( 2a) =-; a+l •

6. Let us now exclude all forms which have one of the quantics as a factor, and seek for the generating function of the perpetu1ints that are left. We shall· ca.II this the red need generating function G~. Its vn.lue is

,G;.

= G,.-rG,._1+ (r) t . G,-i+ . .. 2 G,-z- . .. +(-) (r) 1

.

+(-y- 1 G1 +.-s+er can never be negative; it can, however, be zero. Let r rep1·esent the sum of those terms of the series we are discussing, for which r-X-s+er 0. 'l'hen

=

r-3 ~ t=O

(-)' ( r

t

=

)

G,._,(l-x)2r-5_r

~ (-)''+s+1.c"·2,-2"(.s-er)[(r)(' _·-2)n -·• 2 -~-(J' •

w, ~.

(J'

fT

/IJ

,1', ..

r-1) (r-2-s+er)Bw-s, 1+ ( s-er r ) (r-1-.s+er) Bw-s,o] +r ( s-er 2

r)

__ "_, ( - )•·+s+l X w2s-20-(s-er) [( 2 (r-2) U S-u {B w- .-:,2 +2n 111-s,l +B-m-s,Oft r-1) {Bw-.,,1+B,r-,,of + ( .~-er r ) B,,,-.,,o] - r ( s-er

• For

%(

8 -;") ( : : ; )

2•-~• = ( (z')) (1 + 2:t + z~)"- 2•

295

[Dec. 14,

182

Now the limits of suuunatiou for s are O and w.

Then we have

i (-)' (2r-4) = (-)'" (~r-5), S

•=0

W

~(-)' (2") (w-s+8) w-s+2 8 2 s

Hence

i (-)

r

3

t=O

1 (

r ) G,._i(l-x) 21·-

t

5

-r

= ~(-;r+io1+lxw[ (2\~ 5) {( ;)-( ;) +1}-(:·=t)]

=-(-t(l-x)l!r-5[(1' 2 1)+xJ. Hence we find where

a:. (l-x)

r is the sum of terms from

2r- 5

=r

(A) for which

r-A-.~+"' = 0.

Since

and neither of these latter quantities, indicated by the brackets, can be negative, they must separately vanish. Thus

(I')

~( - )" 2•·-A-cr (r-A) 1, -- .., tr A 1·~.Jw-r+A-o-,>i..X .,• Now if

A > 2a,

B .., A

= 0,

thu1-; we may limit the series to A :t> 2(w-r+A-IT)

or

A 2.

And since

(efg)

= ¼{efg} '(efg),

Ta..,a... ......,.S

= 0,

unless h > 2. That is, we need only consider those terms of the sub8. stitutional series for wl:iich h Let us suppose that a2 > 1, then every term of Ta.,. oi, ...• a.A hns a factor of the form {efg} ' {hi} '. But {efg}' {hi}' S is obviously zero unless S is of the form (efh)g .. i-r,, ..•, but even here we have by the fundamental identity·

=

{efg t' {hi}' (ejh)g.,i.,

= ·! efg l' 1hi l'(ejg)h,,iz = 0.

'fhus we see that the substitutional theorem gives S

= An-2,

l, I

T11-2,

I, l

S,

for every other term vanishes. Now, T,.-2, 1, 1 is a sum of terms each of which has a8 a factor a positive symmetric group containing n-2 letters-that is, nil the letters hut two. This group may then be defined by the missing letters; if these bee and/ we may write it [ef]. Then it i:; clear that [ef]S

= 0,

unleHs both e and f appear in the determinant factor of S. Now, if Q be n sum of terms such as S, the necessary and sufficient condition for the identity Q 0 to be true is that [ef] Q 0 for every choice of the letters e and /. It is necessary, for if Q 0, obviously [ ef] Q 0 ; it is sufficient because

=

=

=

=

Q

= LA:l'Q = A11-2,

1, 11' .. -2, 1, 1

Q = A,,-2, 1, 1 !H[ef] Q

where H is a substitutiona.l expression. Let us apply this to the proposed identity,

unless both e and fare found amongst the letters a 1 ••• a ,b1 •• • b5 • Hence, operating on (VII) with [a>. b"J, we find

313

200

[Dec, 14,

TERNARY PERPETUANTS

for .X = 1, 2, ... , r, condition that

µ

= 1, 2, .. . , s.

[a 1 a2 •• • arb1 b2 •• • b,]

But this set of equations is the

= [a1a2 . .. ar]+[b1b2••· b,],

that is, a sum of two sets of terms, one of which involves the a's only in the determinant factor, the other the b's only, as we proceed to show. In the determinant factors (a>.a,..b.,) and (a>.b,..b.) we may introduce the letter a 1 by means of the fundamental identity. Let us suppose this to have been done in the first place. Then, using the notation of (VIII), we see that EA.,.; • is zero unless ;\ or µ is unity, and that F,. ; ,.., ., is zero unless;\= 1. Operating with [a1,.b,..], we find

'i:.EA,o:; ,.-'£.FA ; ,.,.,= 0 . ., " F>- , ,..,.,=0

But

'l:.E>-,z,,..=0 "

unless ;\ = 1 ; hence unless;\= 1. Again, if ;\ =I= 1, hence all the E's are zero. Now use the identity (a 1 b,.. b,)

= (a b b.)-(a b b,..) + (b b,.. b,), 1 1

1 1

1

and we find as beforn that all terms (ab,. b,) disappear, and hence

[a 1 a~ ... arb 1 b2 ••• b.]

= ~A

1, ~.

s (a 1 a 2 a 3)+}:B1, 2, s (b 1 b2 ba).

'fhus any relation of the form (VII) that can be obtained is incapable of giving syzygies, as it only yields obvious identities between products of forms. 'fhe argument applies in the same way to quaternary forms and to forms with any higher number of variables.

314

X. The Linem· Invariants of Ten Quaterna·ry Quadric.~

By Mr H. W.

TuRNRULL,

Trinity College, and Dr

ALFRED YouNG,

Clare College

[Received 4 l\fay, /lead 2G Jnly, 1926.) INTRODUCTION. In the year 1825 a problem was proposed by l'Academie de Bruxelles which has never been satisfactorily answered, although many attempts have been made. In the original form• the problem was to find the general property holding between ten points on a surface of the second degree. Soon afterwards* the question was modified, and required the three dimensional analogue of Pascal',;, theorem for conics. As Chasles remarks, it is far easier to answer the latter question than the former: in fact he gives in Note XXXII various results which meet the case. So also have several t ot,her geometers. From the outset Chasles perceived the fundamental importance of the first question for the future progress of the theory of quadric surfaces. He observed that for conics, Pascal's theorem has two e!'scntial aspects, giving (I) The relation between six arbitrary points on a conic. (2) 'fhe relation between an arbitrary triangle and a conic. Hence, in space, there will presumably be (1) The relation between ten arbitrary points on a quadric surface. (2) 'l.'he relation between an arbit,rnry tetrahedron and a quadric surface. For the conic these two relations are practically the same but for the quadric surface they are essentially distinct. 'fhus relation (2) is readily found, whereas (I) presents great difficulties. In fact (2) can be quotedt as follows, from Note XXXIl of Chasles' .Aperpu : Let six edges of an ai·bitrary tetrahedron meet a quadric S'ltrface in twelve z>oints, lying by threes on four planes, .mch that each plane contains three of the points lying re.~pectively on three edges issuing from a vertex of the tetrahedron. Let each plane be associated with the corresponding oppo.~ite plane of the tetra./,edron, cutting it in a line. 1'/ten the four lines .~o found are generators of the s0;me system of a quadric. A geometrical proof follows directly by applying Pascal's theorem to the six points on three coplanar edges of the tetrahedron. For its Pascal line cuts each of the four lines in question, so that the fonr line's have at least four tram1versals. But relation ( 1) is something very different, and the comment of Chasles is noteworthy: • Cf. ClmRleR, Apert;u Jliwtorique des mi:tluule, en Gl!u111etrie (Pnrie, 1875) (reprinted from the memoir in the Mem. COltrUIIIICeR ]!..4 + µ.6, .. ............. ... .. .............. .......... .(8)

G = - >..•µ.• [(1234)(6891)-(1236)(4891)) (2346)(4689)(8923). This certainly does not vanish identically when the points 1, 2, 3, 4, 6, 8, 9 are independent, for it does not vanish when 1689 are coplanar. But expression (7) vanishes if for symbol l we substitute any of the following : 0, 2, 3, 4 5, 6, 8, 9, >..8 + µ9. Regarding l as current coordinates, we infer that the equation

G=O is that of a quadric through the points 0, 2, 3, 4, 5, 6 and the line (89). Hence G and A0 , which are of the same degree in each symbol for point or line, can only differ by a numerical factor. In fact G = -A 0 , . .. . ... ....... ... . ... . . . . .. .. . .. ...... .... ... . . . (9) as is seen by comparing coefficients of

a.2 b,b, c,c, d,d, e,' JJ. g.g, on both sides of (8) written in foll, namely

323

274

MR

a.',

b,•, Ci',

d,•,

TURNBULL AND

I (abed) (aefg)(bjhk) (gchk) (dehk) a1a2,

a 1 a.a,

b,b,, c,c., d,d,,

b,b,,

a1a.,, b,b.,

C1C3,

C1C4,

d,d,,

D1t

a/,

bf, c'I.', df,

YOUNU, THE LINEAR a~aa,

a,lQh

b,b,,,

b,b.,

C::Ca,

C:!C,"

d,d,, d,d,,. d,d., e/·, e,e,. , e::::, e1 e"J., c1 ea, e:.:t':1, e::e.a, f,f,, fl, J.J. . /i", J,f•. f,J., j,f,. g,•, g,9•• g,g,, g,g•• g?, g,g•• g.9,, I, • h,•, h,h 2 , h,h,, h,h., h,h,, h,h., ' k,k,, k,k., k.2, k,k,, k,k,, k.2, k,k,, 2h,k,, h,k,+h,k., h,k3 +h,l,,, h,k,+h,k., 2h,k2 , h,k,+h,k,, h,k,+h,.lc.J,

.

a:/, bf,

cl,

aaa,h

af

b,.b.,

bf

C:1 C,,

c,..fl'; accordingly {ab c d e f g h}' I,= 8 ! >..fl'.. . ...... .. ... .. ..•.................. (9) But K consists entirely of 240 terms like I, and is such that {0lf!34567)' 1( = 8 !l(. . .. .. . ... .. ... ... .. . .... ... .. .. ..... (10) It is therefore a multiple,µ say, of fl' and we may write H = f(f!3456)) {789)' (0123) (0456) (1578)(6289)(3497), K = {l ~(07)-(17)-(117)-(37)-(47)-(57)-(67)) 11 ................. .(11)

l

=µfl.

The following special case shows thatµ= - 20, which finally gives fl'= -.JuK. . ...................... ......... ........ ..(12) Calculation of the numerical coefficient µ.

Let the symbols of ten qua24) -{48) (..(a&, def, ghij) and the interchange of any two of g, h, i, j merely changes the sign. Consider then rli (7140) [a/3) (714, 0a/3, ,y&t), where a, /3, 'Y, S, e, t are the digits 2, 3, 5, 6, 8, 9 in some order. These fonns arc unaltered by the operations of (g,). It follows that the form is zero when a/3 is any one of the pairs (63), (82), (85), (92), (95) (from the last two operations of g3), or of the pairs (89) or (52). Now rN {afJ'YI PI.= o, thus rN {7140) [183) (714, 083, 2569) + {86) (ffi, 086, 2539) + {63) (ffi. 063, 2589)) = o.

344

295

INVARIANTS OF TEN QUATERNARY QUADRICS

The third term is known to be zero, and the first two are equal because (g,) contains -(63)(25), and hence they are zero. In this way we see that every form is zero for which the first group of Pis {71401. Hence rT•. ., ,. ,. ,. , 1. = 0. As regards T•. •• •• ,. 1 1 0 we may proceed as before to shew that 1'10 = Ix N [abed) {efj fgh I (d i11 rno:1 iu the Hitlllt' place, "fjher die charakteristischen Einheiten der i;ymmetrischen Gruppe", Herr Probenius referred to my two papers on substitutional analyi;is, and expounded very clearly the connexion between their contents and his own work. In view of his exposit.ion there iH no O('\'asion for me to go over the Harne gronnd. But l fret that it is ne(·t•ssary to ,-;ay a word in justifieatio11 of rny not adopting Iii:-- notation tl1n111ghout 1ny ,rnrk. 'l1l1c ju;;tificatio11 lies iu my purpose of atten1pting to solve substitutional equations; a11d for this purpose it seems to me that it is necessary to w,c a 11otation which exhibits the relative poi;itions of the letterH. l•'or the purpose of the general theory of groups it 1s different. [ndced, so far as that is '.

~

r=l

ArPrurN,

where Pr is derived from a standard tableau. CoROJ,J,.-\HY.

'l'he i;ame argument is applicable to prove that

NP= NP= PN

>.

~

r=I

>.

~

r=l

A,NurPr; ArNr.

= ~ ArP ii. is well known that if the order of ,p(x) is less than h-1 "', 'P(fr)

~ /'(tr) X

Hence

[,..

Now

= O.

= _ ~ c i(fr)-J' (fr)+½/" (fr) ~T

j'(fr)

=~fr-(!)= ~ar = n. 'l'hus the induction is proved when no two of the a's are equal and a,,> 1. When ar ar+I the letter a,. cannot occur in the a,-th row of a standard tableau at all . But in this case the corresponding term in the sum X contains a factor a,- l-a,+1 1 which is zero; so that the proof is unaffected, and we may remove the restriction that two a's must not be equal. And when a,. = l, the term in X for r = h is exactly that given by putting a,.= l. The formula is obviously true for n 1, 2, 3, 4, as can easily be seen by writing down the standard tableaux in these cases. Hence it is always true. We shall call this number Jai. ..,, ... ,..,,, and drop the suffixes, when no confusion can arii;e; it is the number which Frobeni11s calls f or x11 • Moreover, in the series

=

+

=

for which the coeflicients were obtained in my second paper,

- (L)2

A- n.' . 8. 'l'he proof given for the above theorem establishes, m fact, the useful formula

1t is interesting and useful to point out that / .. ,..... .. ,•A is the coefficient of xj' + 1•- 1 x; + 1•- 2 ••• x:A in the expansion of 0

• E.g. see BurnEid.e and Panton, Tluwry of equations, 1 (1899), 172.

358

A. YotJi\1 precedes at>,. Then, since F, is a standard tableau, at>, precedes a\'/ v unless v < t. Thus the letter at>, must appear in one of the first t-1 columns of F,; and N, has a factor {at\ at>v}' where v < t; and hence N,Pr 0.* The argument is the same when the first rows of Fr, F, are identical and the distinction between them arises first in a later row. s, we know that NrP,. is not zero; and when r > s this vVhen r equation does not always hold good. In all cases N 1 P,. 0, unless r f, for otherwise r < f. Let UR consider Nr, N 1_ 1, • • • in the reverse order, and Id N • be the first N we s. Also we know that P.. contains a sulrntitution -r.,. which transforms Nr into N, t, so that

N. T,r

= T,,.·N,..

It is possible that tliere is more than one value of r for which the above inequality is true. Let these be r 1 , r 2 , • • •• Let us write

N~

= N,(1-T,r -T,r, 1

... )

Then for all values of r other than s . For, by hypothesis, N,Pr is zero when

t is lei:;s than s a.ud not equal to r, and hence N .:P,.,

= N,P,.,-Tsr,Nr,Pr, = 0.

Further We call N; a prepared form. Proceeding, we take the next value of s for which there exists a .different number r such that N,Pr is not zero. We then replace N, by

• For P, has a factor {ar the operations of the gro11p as n·o111pll•X 11nitH, t else as 111atrin•,s. tlii11kinµ- of tl1t• rc,.;ults as rderring- to 111atrix gro11ps. l•'roh1•ni11s lirst ("011silll'rS tlw k different dasses into whi1·h tlw cI,,_ mcnts of a gro11p H 111a\' lw 1liYi1lt•,]. all the clements conjugate to a gin!11

=

• Q.S.A., I, 141-143.

387

l!J27 . )

:Wl

ON (llJAN'l'J'J'.-\TJ \' E :,;t ·i;:-;TJTI ·noNAL ANAI,YSIS

de11w11t fon11i11g one clas:,;. 'l'hc 111111ilH'r of clements belouging to a class is calle,rr, .,N,M.• , we find the conclition expressed by k conditions

'l'lrns tl1e determinants 1T are tho factors of I. Now Frohenins hns shown tltat the dotienninant J has k irreducible factors, viz. tho group ddenniu:wts of tlw l,; irreducible linear substitution groups. 'l'lurn tho expressio11s JI are tho gro11p dotermi11n.11ts of the irreducible groups. Ono of the cletenui11a11l.,; 11 ca11110L he a pro1l11ct of two of the gronp dctcnnina11ts, for the s111u of the scprnrns o[ tlte clegree,; is n! in c:ich case. 'l'h11s [L,1 a'.'.,x1,1] is tho grcrnp 11rntrix of an inedncihle represe11tatio11 of the symmeLric group as n. li1wa1· s11ltstilutio11 group. Nol\· lot b;'., he Lite coetlicic!llt of the permutation Qin thn expansion Clf f' ,.rr,, N ,M.,, :-;o llt:tt

P, rr,, N , M, 'l'he11, si11co we li11d tl1at

(.!

=

P,.rr,., N,AI,

= Ll,'.2.r;.

L a'JJ 1Krr.,..N""Jll",

•• A

= L ~ b~Jt~,..P.rr.A.N,..Jll,... '} K,

A

(IY)

Honco • Q.S.A. III, § 7, pp. 267-269.

398

(II I)

I,!

[Nov . l l,

250

1

uulei:;s "

=r

and ;\

= s, and a;>. belong:-; to tho same T as b?,,; "t-b~,at r . Now N, must contain a permutation JI,., for which J1,P, J1; 1 Prt. 1 Similarly 1\ contains a permutation ur., for which oT,N, "fl!,N,.

=

=

rr,. = ur,.JI,. and rr,,. = J1,rn-,.

Hence

Now J1,:- 1 tu,.J1, is a. permutation of

Hence 'fhis permutation tu; is the inverse of tu. above, since

'l'hus, slightly changing the notation, we have, in this case,

rr, .•

= Jl,.tu.,

(I)

where J1,. tu, J1,:- 1 helougs to P,. n.ncl tu.; 1 J1,.tu., to N.,. Hence when N,.P, is not zero it is eqnal to ±N,rr,..,P., according as JI,. iH even 01· odd . And, so far as P., alone is concerned, 1-ur,. Ill,.

=

It may lrnppen that N.P, =I=- 0 where t 'l'hen (Q.S.A. III)

N,. Ill,.

>

s.

= N ,. -ur, N, (l -r.r1)

In fact 'l'hus, when the values of s for which N,.P,=l=-0 are ri, r2 ,

.•• ;

the values of s for which N,.,P. =I=- 0 • Q.S.A. III , § 4 ; Q.S.A.J., p. 15. J Q.S.A. II, 363, § 13.

400

are

[Nov . H.

A. YouNc:

i58

r"''

f"t,. 1 ,

etc., and so on ; then

111,. and here

= 1- ~m-•., +~m-.., m-,.,,. r

< r"
.-i L>., and u,, ~ .~ > a>.. Let s lie betwoeu aP ➔ 1 a11d up, i.e.

'l'hen the trnn;;poi-;ition (a,,.,:i:) appears in the factor J,>.,p• \Ve have seen that auy term which hn.s a factor (a.,,x) to the left of this, where r ~ rT, eitl1er is zero or would re11wve the transposition (a,.,x). We may go further a11-+;\-p-l

[1-

a/>_1-ap

a,,.1-.+)\-p

J...[1-

(tlw factors i11 this pro'111ct ari~ing severally from _

I

-

(.,p, I,>.,p-1, ... , l,>.,r)

-[(aA->.)-(«e-p- 1)-1] (a.., -;\)-(a,,-1 -p)

· ··

[ (a>.-A)-(a,. 1 -r-~)(a.., -,\)-{a,,- r-- 1)

= [J-

l]

(a>.-,\)-:a,,-p-1)] [ l - (a>.-,\)~(a,,-1 -p)] ···

1 (a>- -X)-(u,-r- If

Let us turn to the case where A ~ r, and the factor (a."x) appears in 'l'he argument is very similar; this transposition appears in

J(>-J(M.

410

A.

268

[Nov. 14,

YOUNG

K>-.,., but there arn transpositions (a..,,x) which mn.y appear in the factors J(>-,,•+i, K>-,r+ 2 , ••• , K>-., p, which hn.ve to be taken into account owing to the

equation

= - P(a,.,x) N.

P(a.,,x)(a,.~x) N

No other transposition from the r-p factors cn.n appear in n. reduced product. And thus the coefficient associated with the trnnsposition is of exactly the same form as before. 'rhe vA.lne of the coefficient depends entirely on the transposition, and not on whether it appears in a. I{ 01· an L factor. Consider then any reduced product. The different transpositions have coefficients which are eo.ch derived from a certain set of the factors K>-.,,., L>-.,,., and these sets cannot overlap, so that the actual coefficient of the whole is expressed as a sum of terms each of which is of the form

Ed

where F(x) is a function all of whose zeros are different and equal to some or all of u,-r-1, r 1, 2, ... , h. 'l'he sum of all such terms from ;.\ = 1 to ;\ = 1 is zero by (lI), § 7. We have oufy then to calculate the numerical coefficient when z 1s absent. '!'his is

=

h+

n. well known result*. 'l'hus the theorem is proved.

9. '!'hat the factors H and L are essentially the same except for the sign of the transpositions may he seen hy considering the case where a>-.-1 a,. 'l'lw ,\-th term i1-1 absent. 'l'he µ-th term(µ..-1,tX)

]

•

11>.-1,

(aA, 1:c).(aA-1 , ,,:r) J>,,. N,.

=

Ol

(a>-.-1, ,.x)

L (a>-., 1.-1 . ") P,,.N,,. =

1-1

-(a>-.-1, ux)P,,. N,,.,

• Sec, for example, Q.S.A., § 3 (2).

411

H)29.]

ON QUANTITATIVE SUBSTITUTIONAL ANALYSIS

269

and hence

1(,.,,J(,.,A-1 P,.N,.

in formal analogy to the L faetors.

10. Consider a stn.ndard tableau F,. of T,.i. .. ~ ....... ,. of n letters. If the last letter a,. be removed a standard tableau Ji';.. of n-1 letters is obtained. Moreover, when N,P., is not zero and a,, lies in the same row in F,. and F.,, N:.. p;. is not zero. And in the same case when N, P, is zero, p;. is also zero. For otherwise the vanishing would he cansrnl hy a letter ae which is in the same column with a,, in F,. being in the same row with it in F,. But the change from F,. to P., can he made hy a column permutation followed by a row permutation ; the column permutation wonlJ>. Jn~, [_'/'... /l-t + -1.

J

111,:!~

'l'he tn1.11svositio11 (a,, ..1 a,,) gives rise to the matrix 111~1 ; this trnnsposition invariably changes a standard tn.blean with a,, above and a,,_ 1 below into a standard tableau with a,._1 above and a,. below; and 111 21 E .. -1,/l-• •

=

414

0N

272

=

QUANTITATIVE SUBSTITl?l'lONAl, ANALYSIS

=

Ma, fl-I; m 12 0, for none of tho transpositions of m11 from the upper to the lower row; and ni~·l = J\l.. -1, fl•

I(

moves n.n a,._1

!tf

0 ] -[llfa.,fl-1 Ea.-1,fl-l Jlfa.-1,fl • -afl -

Thus In particular, Ia. ..

=E,

M a,a.-1

M.., 1

and

=

[ EIlla. •a.-2

a.-1, a.-1

J ,

= [1 1 ... 1],

a. matrix of one row and a-2 columns. In the case of Ta., 1, 1, ... , 1, called by Frobenius the case of unit rank, K 1 PN= [1+

a+!-l (bhx)J[1+ u+!_ 2 (b1,-1x)J ..

[1+ a~! (b,x)Jff a, . a.x}

Then wo may dm~igrrnle tho trn.nsfon11ing matrix as

[E, a +:-l

I (o,l1) -_

l

E,

0

where Also where

'111(a.,hJ] ,

_ (a+h-3) _ (a+h-8) h-1 . h-2 ' s -

r-

-

·mah-

0

c-11l(a.,h-l)

(• >

t -_

Ee

m( .. -1,hJ

]

•

(a+h-4) h-2 . 415

ON QUANTil'A'flYE 8UBS'l'I'l'U'l'IONAL ANALYSIS

(FijtJ,, Paper)

Bu

ALFRED

YouNG

[Received 30 August, 1029.-Read 14 November, 1029.)

For linear substitution groups of degree two the rotation groups of the regular solids in three djmensions are of fundamental importance. It is natural to helieve that the rotation groups of the regnlar solids iu Euclidean space of any nnmher of dimensions must occupy a prominent place in the theol'y of linear substitution grnups of higher degrees. It is well knowll tlut in space of n dimensions when n > 4 there are only three regnlar !:ioli,ls, tlie simplex, the figure corresponding to the cuhe which we may call the hypercube, and that which corresponds to the octohedron, which Wt\ c,tll here the hyper-octohedron. 'l'he simplex in u dimensions has n l vertices, the symmetric permutation group of its vertices is practically the group which includes hoth the rotations and the reflections of the ligure into itself ; the alternating group is the pure rotation group. 'l'hti exte11tle.. t,

where t is the sum of the memherR of a conjugate set, o.nd the coefficients A are real numbers. Now the inverse of o.uy pernrntation belongs to the so.me conjugate set o.s itself, and hence the square of such an expression o.s (~;>d) cannot be zero when the ;\'s are all real. Hence the s of the permutations of this set contained in y Pa.1•-i••·•,.·

To find this we use a function Wy, which is merely what W11 becomes on change of notation, and observe that, since the equal cycles are really equivalent, we must divide B! by Il y, ! ; and, since the letters of the different groups of Pare really distinct, we must multiply by Ilar ! ; and finally, since the letters of each cycle may be permuted cyclically without changing the cycle, we must divide by IlrY,. Hence

and therefore

n y BY.. =

(a

n

1 a 2 •••

a 1,

) n-,,., =0

unless

~+l ~JLr ~Ar,

B>-,,., =l

if

~+l > JLr>~,

B"=~ ,,..

if

Ar= P,r,

B;, = 1-~+l if

Ar+l = P,r•

Since the representation of the group is irreducible, there is only one invariant quadratic. It is therefore sufficient to show that this quadratic is invariant for the group. The symmetric group of degree n is generated by the n- l transpositions of consecutive letters (arar+l) ; therefore we need consider only these transpositions. Consider the result of operating with (arar+1) on the variable x>.,>-., ... >.k• There are three cases: (i) Neither r nor r+ I appear among the suffixes of x; the variable is unchanged. (ii) Both r and r+ 1 appear among the suffixes of x; the variable is changed in sign. (iii) One, say r, but not r+ I appears amongst the suffixes; then x is changed into. another variable y in which the suffix r is replaced by r+I.

438

]!ml . ]

ON

QUAN'f11'A'I'JVJ-: SllBSl'JTli'J'IONAL ANALYSTR

Let us now consider the effect of this transposition on X,._ 1 ,._ 2 ___ ,. 4 • distinguish the same three cases :

203 We

(i) Neither r nor r+ I appears among the suffixes. Then, these suffixes being arranged in ascending order of magnitude, r and r+ 1 must lie between a consecutive pair, say A8 and .:\,+1· Then the coefficient of x"• ·•• /1.k is zero when r or r+ I appears amongst its suffixes, unless only one appears and that as ,-,,.. Moreover, when ,-,,. = r or r+ I, B~, = I. Thus the linear function X is unaltered by the transformation. (ii) When both r and r+ I appear among the suffixes of X, we may take .:\. = r, .:\,+1 = r I. Let Y be the transformed form of X.

+

Then the coefficient of x"' ... is zero unless ,-,,. = r or r+ I ; and thus, when only one of r, r+ I appears among the suffixes of x, it is µ,,. Now for µ,. = r, B~. = r; and for ,-,,. = r+ I, B~. = -r. Moreover, when both r and r+ I appear among the suffixes of x, the sign of xis changed by the transformation. Thus

Y=-X . (iii) We shall call this for brevity Xr, and associate with it the function in which all the .:\'s are the same except A8 = r+I, and call that Xr+1· We want to find the transformed functions Yr and Yr+1• The coefficient of x,. is zero in both cases when r or r+ I appears as any suffix except,-,,. or µ,,_ 1 • When neither r nor r+ I appears in the suffixes of x we have a sum of terms ZA,.x,., the same for both Xr and Xr+t• and also for Yr and Yr+1· In Xr we have the following coefficients:

µ,=r, B,., =r ; µ,=r+I, B,.,=I ; µ,,_ 1 =r, B,.,_ 1 =-(r-1). In Xr+1 we have the following coefficients:

,-,,. = r+ 1, B,., = r+ 1; 1-'a-i

=

µ,,_ 1 = r, B,.,_ 1 = 1;

r+l, B11.,-, = -r.

Then, if y is a variable for which µ,_ 1 = r, ,-,,. = r+ I, y will appear

439

A.

204

[June 18,

YOUNG

in Xr with coefficient -(r-1) 0, and in X,+1 with coefficient (r+l) 0. Let ur be a variable x in which µ,. = r, but r+ 1 is not among the suffixes and ur+1 the same when µ,, is changed to r+ 1 ; let vr, vr+l be similar variables when r or r+ 1 is in the position µ,,_ 1 • Then

and hence

and

Now in the sum H, Xr.and Xr+1 appear in the form X 2 1 1 g [ r(r-1) r + r(r+l)

x2 ] - [ r+l -g

1 y 2 1 r(r-1) r + r(r+l)

y2 ] r+l '

where g is a constant. Hence H is invariant for the transformation (arar+l). We have yet to consider the transformation (a 1 a 2 ). The only variable affected is x 2 ,,., •.. ,.., and this variable occurs only in X 2,A 2 • •• >--.- Thus every X,. which has not 2 for its first suffix is unchanged by the transformation. Now x 2 ,,., ••• becomes on transformation -x2,,., ... -x3,,., .. . • Let X 2, ,., ••• become Y . In Y there may at first appear variables x,,,, ...,,. which have zero coefficients in X. Let us consider the possibilities, as follows: (i) Two imffixes lie between .\, and .\,+1; this variable arises from two terms of X in which one or other of the suffixes is replaced by 2. They will appear with the same coefficient but a different sign, and their sum is zero. (ii) Next we may have three v's between .\,_1 and being.\,; say v,_1 , .\,, v,+1·

440

,\a+l•

one of them

Hl31.J

0N

QUAN'l'l'fA'l'lVE SUBS'l'l'l'U'I'IONAJ, ANAl,YSIS

205

This term arises from three terms of X, each with µ.. 1 = 2; they are µ..,_ 1 = v,_1, µ..,=A,, coefficient ±2A, 0, µ..,_ 1 =

v,_11 µ.., = v,, coefficient =f 20,

µ..,_ 1 = A,,

µ.., =

v,,

coefficient =f2(A,-l) 0,

and again the sum is zero. We may treat the case of four v's between A,_ 2 and Aa+i, viz. v,_ 2 , A,_1 , A,, v,+1, in the same way; the sum again is zero. Proceeding thus we find that the coefficients of all variables which do not appear in X are zero in Y. As regards those terms x,,. 1 ,,., ... ,,.1 whose coefficients are not zero in X, we have: (i) µ.. 1 = 2, the term is merely changed in sign. (ii) µ.. 1

< A2 • B,,_ = 1

1.

The coefficient is O in X.

In Y it also arises from x 2 ,,,., ... ,,_4 and from no other term, and this gives a coefficient - 20; and thus finally the coefficient is -0.

In Y this arises by replacing either µ.. 1 or µ.. 2 by 2, giving coefficients -0, A2 0; in X the coefficient was -(A 2 -l) 0, in Y the final coefficient is (A 2 -l) 0, a mere change of sign. Proceeding thus we find that Y=-X. Thus H is also invariant for (a 1 a 2 ) and therefore for the whole group. is therefore the quadratic invariant.

It

5. We may uow take X>., ... >. 4 as new variables; in this case the qua.], '). = h, It- I, . .. , I, down the leading diagonal; where [T>-] is the seminormal matrix for S. Ll>- T . 0 , 0 , •• •

It should be noticed that the values of,\ are given in descending order, as according to the usual sequence, tableaux with a.,. in a given row precede those with an in a row higher up. In this particular the enunciation of Theorem IV in Q.S.A. IV is at fault. 7. For the sake of clearness the matrices hitherto used for T corresponding to the units Prar,N,M, will be called the natural matrices, and these units the natural units. It has been shown that there is a matrix M of special character, by means of which we can pass from the natural matrix Oto the seminormal matrix D of a given expression, that in fact D . M-1 CJ!. 01 ... 01,

\Ve call M the seminormal transforming matrix for '11 a,,• Consider now an expression S involving only the first n-1 letters. We have seen that, if [T>.] is the natural matrix and [T/] the scminormal matrix for LlA Ta, ... Q/,, then the natural matrix for Ta, ... a,, is 0 , ...

s.

:l s.

['1',.]

I[o 0

...

0

[T,,_iJ

0

...

...

J-1

=

IKJ-1;

aud the scmiuormal matrix is

···i ' 445

A.

210

YOUNG

[June 18,

J-ll(J,

but this is equal to where J is the matrix

M,. 0

0

O

M,._ 1

0

0

0

MA being the seminormal transforming matrix for

We notice that Smay be the perfectly general substitutional expression of the first n-1 letters, in which case [TA] is a group matrix, and, as Schur has proved, is permutable with no other matrix except E . Then, since the matrices [TA] are all different, the only matrix permutable with K is L, where L is the matrix obtained from J by replacing each M,. by p,. E,., p,. being an arbitrary constant, and E,. the unit matrix of appropriate degree. Then, since M-1 /KJ-lM = J- 1 KJ,

K is permutable with J-1 M J-1 , and J-lMJ-1 = L .

But both I and M J-1 are matrices with units in the leading diagonal and zeros below it, hence in L every Pr is unity, so that

L=E, and therefore

M=IJ.

'fm:01rnN Ill. The seminorrnal transforming matrix for T being M, and that for~-' T being MA , then

010 , •• • 0 ,,

=

T

M=IJ

where I is the transforming matrix corresponding to the addition of tlie last letter in T, and J is that matrix /orme,d by the matrices MA in the leading diagonal, with zeros elsewher~,

446

H.lal.J

ON (lUAN'l'l'l'A'l'lVE l:WllS'l'l'l' U'f lOl\,\I, ANALYSIS

211

8. Consider the matrix for (a,._ 1 a 11 ) in the seminormal form. We first subdivide the seminormal matrix according to the positions of the last two letters. When we take the position of the last letter only we have a system of subdivision according to the sequence of tableaux for

a,, T,

t:,."_1 T,

.•• ,

a1 T.

Taking a further subdivision we use the tableaux for

in this sequence. By Theorem II the most general substitutional expression invol~ing the first n-2 letters only is represented by the appropriate group matrices [T""] down the leading diagonal and zeros elsewhere; call this X. The transposition (an-l an) is permutable with all such expressions. Now Schur* has shown that the only matrices permutablo with X are those which are made up of quadratic submatrices of the form

occupying the rows and columns defined by the tableaux for t:,.A t:,." T and t:,."t:,.>.. T: submatrices qEAA occupying the position on the leading diagonal of [T>.>..] in X, and zeros everywhere else. The seminormal matrix for (an-la,.) is then of this character. It may be noticed that there are obviously more independent matrices than those appropriate to A+B(a,,_ 1 a,,) of thi8 nature. Let r,,_ 2 1,e an operation which represents taking the sum of all the (n-2)! permutations of the symmetric group of the first n-2 letters in the following expression. Then r.,_ 2 (a 1 a,1_ 1 ), or ru~(a 1 a,,), or in fact r 11_ 2 S, where S is any substitutional expression, is of this nature. I.I. Consider now the transposition (a,._ 1 a 11 ) . We fir8t take the case where the row8 of the tableaux for T arc all of unequal length. Then the natural matrix for (a11 _ 1 a 11 ) consi8ts of quadratic matrices of the form

[~ ~], and of matrices E along the leading diagonal and zero elsewhere. • Bcrlir,,er Sitz1mgsberichte, 1905, 406-432 (§3).

447

A.

21i

[June 18,

YOUNG

To make this clear we give an example, the natural matrix for (a6 a 6 ) in TuThe matrix has nine rows and columns, but since we are concerned only with the last two letters we can group together the first four letters. Arranging the tableaux according to their proper sequence, t.hey will be: T, (I row),

T 3, 1 (3 rows),

T3,1 (3 rows),

T 2, 2 (2 rows) ;

and the required matrix is

[f'

0

0

0

0

Ea1

0

Ea1

0

0

0

0

E22

As a second example, (a6 a 6 ) in T 321 is 0

0

Ea1

0

0

0

0

0

0

0

E22

0

Ea1

0

0

0

0

0

0

0

0

0

0

E211

0

E22

0

0

0

0

0

0

0

E211

0

0

Let O be the natural and D the seminormal matrix for (an-l an), then

(IV)

OM=MD.

The matrices will be 1mbdividod as in § 8, so that M= [m,.].

Let r refer to the row defined by AA 2 T ; then in the matrix on the left of (IV) the only elements in the r-th row are mr,, the only elements on the r-th column in the right-hand matrix aremqrDrr, and hence, equating

E,.,.

448

HJ3l.]

ON QlJANTJ1'A1'1VE SIJilSTI'l'UTJONAI. ANALYSIS

218

the elements in the r-th row and column,

or Drr= EA>.• Let r refer to the tableau fl.A fl.,. T and s to the tableau fl.,. fl.AT, where A 3. Then the seminormal matrix for (arar+i) has unity on the leading diagonal and zeros elsewhere on the ~-th row and column; - I on the leading diagonal and zeros elsewhere on the {-th row and column; and the f-th and 11-th rows and columns have the quadratic matrix

where they intersect and zeros elsewhere, where

This is the sum of the projections of the straight line joining ar and ar+l on the first row and the first column of the tableau ; we call it for brevity the axial projection of arar+1· We may then enunciate the result as follows: 'f1moni-::11 IV. The seminormal matrix for (arar+l) has zero everywhere, except (i) unity on tlte lea,ding diagonal wltere tlte tableau ltas ar, ar+l in tlte . same row; (ii) - I on tlte lea,ding diagonal where the tableau ltas ar, ar+l in the same column; (iii) a qua,dratic matrix

the elements being in tlie positions of the intersections of a pair of rows and columns wltose tableaux differ only by the exchange of ar and ar+l-tlte value of p being the reciprocal of the axial projection of arar+l in the tableaux.

12. The seminormal matrix corresponds to the variables defined by (Ill), and the quadratic invariant ~

,\~, (; 2

.., B

r

~r.

To find an orthogonal representation of the group, new variables 1/r must

453

A.

218

[June 18,

YOUNG

be chosen, for which the form I:77r2 is invariant. choosing Y/r =

,\rr

This may be done by

'V Br Sr= Pr Sr• l;

1;

This is equivalent to transformation by a matrix which has zeros everywhere except in the leading diagonal; in other words, the r-th column of the matrices is multiplied by the same number p,, and the r-th row is divided by Pr· In fact, we may write D as the orthogonal matrix where

and M has zeros everywhere except on the leading diagonal, which. has the elements Pi, P2, .. •, P1· The transformation (a,ar+1) for the orthogonal matrix is the same as for the seminormal matrix except that the quadratic matrices which there appear have now the form

Since this matrix must be orthogonal, ro

=

y(l-p2).

'l'm:onEM V. The orthogonal matrix for (ara,+1) ia identical with the aeminonnal matrix except that the quadratic matricea which appear in it are replaced by the matricea [

-p

py(l-p2)]•

y(l-p2)

p having the same meaning aa before.

13. We call the rows and columns which appear in the quadratic matrix in the argument of the last paragraph, the a-th and t-th, a< t. 'l'hen

(VI)

454

1931.]

ON QUANTlTATIVI•: SlJBSTl'l'lJTI()NAL ANALYSIS

219

a. relation which connects the coefficients of the original quadratic invariant § 6, l

H=l: BX/". r

This relation will be required later. 14. Consider the product

where m,11 is a seminormal unit, ands, t give the rows and columns of the last paragraph.

Now

+~ P,u,,N, M,+ ...}. We are not concerned with the values of the constants A 11, . Consider the effect of mult.iplying the term beginning with P" by (a,a,+ 1). Since v < t, the position of an in the tableau F" is either in the same row or lower down than in the tableau F,; if (arar+l) F" is nonstandard, the equations expressing the resulting term in standard units cannot raise the position of an . Thus the only terms which can yield P, or P, are those terms P" which have an in the same row as in F, or F,. We apply the same argument to each letter in turn, and conclude that the only terms commencing with P" which, on multiplication by (a,.ar+1) and then expressing in standard units, contain the terms P, or P1 are those for which F" is the same as F, and F 1 so far as the letters ar+2 , ar+ 3 , •• • , an are concerned. As regards the position of ar, ar+I in F" we may remove these last n-r-1 letters, and consider only tableaux F' of fixed form of r+ l letters. Regarding the possible tableaux F" we see that unless a,., ar+l terminate two consecutive equal rows (arar+1) F., is standard, and so the

455

220

A.

[June 18,

YOUNG

term only yields P. or P, when v =sort. In the case when ar, ar+1 are at the ends of equal rows in F", the pair of equal rows must lie below the arrow in F, otherwise 'V > t; and thus both letters ar, a,+1 lie below this row. After the transposition the standardising equations cannot raise them up to the level of that row without altering the position of Inter letters. Thus

+ l:Au, {P,u,.,N.,M.,+>.,,/ >.u P,u,.,N., M.,+ ..•}, T

giving

>.,,f>.tt = P·

And

+Pl: Au-r {Pe Utr N., M.,+>.,,/Au P, u,, N., M.,+ ... }, T

with the same result. Let us now consider the variables; becomes

the transformation (arar+1)

f,' = -pf,+f,,

changing to the variables Xr of § 6 and using Y, as the transformed variable, we have

Y.= -px,+}•II X,, Y1 = (l-p 2)

. X,+pX

:tt

.

1,

Y,+x, = (l-p2) ~ x.+(l+p)X,. 456

Hl31.]

ON QUANTI'l'ATIVE SUBSTITUTIONAL ANALYSIS

221

If ').,./'>-11 = I+p, then

Y,+x. = Y,+x,, and exactly as was found for the case T k. 1 1 • The full justification for this ratio of ').,. to 'Au will be found later. Meanwhile it should be noticed that, the absolute values of,\,.,. being still unfixed, the coefficients can all be made integral with this ratio. 15. We will now define a numerical function, tho tableau function, by means of the standard tableau. Let F be a standard tableau ; it may bo defined particularly as a matrix

Fs [Yr,,]; Yr.• is the element in the r-th row and the s-th column; in particular, Yrs

is the suffix of the letter which appears in thfa position in tho standard tableau F. The standard condition is y,, > y 1,

when

r > t,

y,. > Yrt

when s > t.

Then a 1 , a 2 , •• • , ai, being, as usual, the numbeni of letters in the first, second, etc., row:,; of F, tho tableau function [F] is the product of a 2 +2a 3 + .. . +(h-l)a,, factors, which arc defined thus. The element y,. gives rise to r-1 factor1:1, one in relation to each row above it. Uon:,;idcr the t-th row (r > t) ; we have a factor Yrst· Wo know that y1, < y,., but y1,. may be either less or greater than Yrs when u > s; it must be greater than Yt,· Let us suppose that y,,. < Yr• and Y"'-1-1 > y,. (when u = a, we shall always regard the non-exi:,;tent Ytu+1 a:,; greater than y,.). Then we define 'Yr,t =

u+r-s-t+ I.

(If it is necessary to specify u more particularly we may call it u 781 .)

Then we define our function as [F]

=

Il y,,,.

(VII)

457

222

A.

YOUNG

[June 18,

We may take the first coefficient in the linear function Xr, i.e. the coefficient of the variable xr, which corresponds to the standard tableau Fr, as the number which we have defined as [Fr], the tableau function of Fr. For consider the transposition (ara,+1) which interchanges the tableaux F. and F,, defined as above; in F 8 the position of ar will be taken as (.\, u), of ar+1 as (µ,, -r); so that y,.., = r, and y,.T = r+ l. Then

In [F,] all the factors are the same as in [F.] with the single exception of y,.T>., which becomes u+µ,--r-A. Now u+µ--r-A is what we have called the axial projection of arar-11 for the tableau, and have written p-1 ; thus [F,]/(l+p) = [F1] .

This is the same as the ratio of the coefficients.\,., Au given in § 14. 16. We now introduce a function which we call the second tableau function. For a given tableau it is defined by the same elements as the first tableau function, and we write it (VIII) Then, with the notation of § 15,

Equation (VI) of§ 13 give1:1

In the expression of § 6 for the quadratic invariant we may then put

458

1~31.]

0N QUAN'l'l'fA'l'l VE SUBSTJTl"'l'IONAl, ANALYSIS

223

It will be noticed that these values agree with the results already found for T -k.i'· Consider the coefficient of xi in H; the only term in which it appears is X ,2 ; thus its coefficient is

Let {31 = k be the number of letters in the first column of the tableaux:,

fJ1 the number in the second, and so on; then the tableau F I is

Then and

., .,

II 'Yrll = (r-t+ l)••;

•=1

II (yr11 -l)

,-1

=

(r-t)•,.

Hence

Let us define H exactly as the sum of the squares of all the terms derived from standard or non-standard tableaux which have different rows, a permutation in the rows only being regarded as not producing a difference; for such a permutation does not alter the corresponding function of the variables x. The number of the terms in H is then

Consider the coefficient of x,2 in H . It is the number of forms PrNr which when expressed in terms of forms with a standard P contain the term P 1• Let PrNr be such a form; then, when a 1 > a 2, an must lie in the

459

224

A.

YOUNG

[June 18,

same place in the first row of Fr; it cannot be raised by tho standardizing equations. Similarly a,._ 1 , a,._ 2 , ••• , an-«,, +«•+1 must all lio in tho first row of Fr- In the same way when a 2 > a 3 the pair of letters an-a.,+"•' an-«,+«.-l must occupy places one in the first and other in tho second row, for unless one is in the first row our equations will not allow either to be raised to it; and unless the other is in the second row it cannot be raised there. If both lie in the first row of Fr, then in the final result P 1 docs not appear at all, or appears twice, once with a and once with a - sign. Proceeding thus we see that Fr, when the letters in each row are arranged according to the fixed sequence, is obtained from F I by a permutation which only interchanges letters in the same column, and each such permutation w.ill result in a linear function in which P 1 appears with the coefficient ±1; and each gives a different term of the sum H. The number of such is

+

VI. The quadratic invariant obtaine,d from the sum of tlte squares obtaine,d from all tableaux with different rows is equal to TnE01tE111

where 17.

TnEoni,;111

VII.

The exact quadratic invariant for T n-k, k is

where When the suffixes of the letters of the lower rows of the tableaux Fp, Fa are .\ 1, .\ 2 , ••• , ,\k; µ, 1 , µ, 2 , ... , µ,k, respectively, the values of the coefficients are

460

IWJI.]

0N QlJANTITATIVE SUllS'l'l'J'lJ'l'JON:\L .-\NAI.YSIS

Bt = 0, 11-1
11-1 > ,\l+u-1•

being reckone,d as greater than n.

In § 6 we saw that the seminormal transforming matrix M is

where of course ,\ar = 0 for r > s. In Theorem III it was proved that M=IJ,

where J is a matrix which may be 1:mbdividcd according to the position of a,.; when thus expressed it contains zeros everywhere except on the leading diagonal. In our case, using suffixes as for 1', and thus writing 1Jf _k,k for M, we have 11

J = [Mn-k,k-1 0

]

Mn-k-1,k

0

.

In Q.S.A. IV, § 12, it was shown that I=

lu-k,k

=

En- k,k-1

I n-2k-j-l

0

E,,._k--1,k

[

K,._A-,kj, 7

with a necessary alteration in nomenclature. An unfortunate error appeared in the formula quoted, 1/(a-/3+2) being given as the coeJiicient of .lJfufl which should be 1/(a-fi+I). Further it was shown that K 1t_k, k _1 , O ] } ln~k= [ • •

l!Jn-k-1, k-1

J(u-k-1> k

We shall use these results to establish our theorem by induction, assuming its truth for all values of n less than the one under consideration. Let

Ka,/H, Ka-1 , {J-A+l • • • [(a-A,{J

= K!"!"tl

= K~, {J-1 Ka-A, {J• 461

A.

226

[June 18.

YOUNG

Then it easily follows by induction that ,. K"-1 K a-1>+1 , 8 K "•fl= a,fl-1

=

[

K~.11-1

0

]

~K~=L11-1

K~-1 . fJ

•

MP -

I KP M o,fl-(a-,8+1)7, o,/1 4-p,fl,

Let

l

then multiplication of these two quadratic matrices yields

MP0,11-

a-,8+2 MP

a- ,8+2 -p

p

... /l-1

(VIII) p-1

a-,8+ 2-p M .. -1,11-1

Whenp=O,

M~.11 =M•,fJ•

Consider now the matrix M,,-k, k· - I J -- [ En-k, k-1 n- 2lk+ 1 K,,-k . k] [M,._k. k-1 M n-k,kEn-k-1, k

0

0

= [Mn-k, k-1 . ZJJ :,-k, k ] . Mu-k-1.k

O

'l'he positions of the elements of this matrix are defined by the positions of a11 in the tableaux; in fact, in the first row they may be defined as

(a,., a,.) , (an, an)

and in the second row as

(an, a,.) , (a,., a,.) .

The assumption that the theorem is known to be true for values of n less than the given value esta.blishes the theorem for the whole of this matrix except the part (.

an,

462

a,.) .

ON

1931.]

QlJAN'J'l'J'A'l'IVP, stmSTITl/l'ION.\T, AN.\LYSlS

l

227

We consider then this last part of the matrix, and subdivide it according to the position of an-I· By VIII ,

[ MLk·· . ,. =

n-2k+2 n-2k+ 1 Mtn - k, k·-1

M'';-k, k

.

1

n- 2k+l Mn-k-1 , k-1 M~-k -1. k Here again the whole matrix is known except the clement M!-k, k in the position ( (

an-1 an,

an-1,

an- 1),

an-1 an)

or, with reference to the original matrix, the position

We keep on repeating this process, and ther_efore

consider in detail the matrix Mf._k (

1;

which may be defined as the element a11-11+1a11-11+2· ··a,.)

a,._·p+1

an-1)

of the matrix Mn-k,k-i· 'l'hc matrices m 11 and Mfi-k, k-l arc defined by tho same tableaux: so far as the first n-p-1 letters are concerned. Now the matrix

where the lower rows of the tableaux F,, Fr are

respectively, the former corresponding to the column tableaux of Ill . Then in these two matrices A1 , A2 , ••• , Ak-l; µ 1 , µ 2 , •• • , l'-k-p-l are tho same.

463

A.

228

[June 18,

YOUNU

Let and the values of the factors b, c being given in the enunciation for values of n lower than the given one. The value of .\., is, in fact, that given for all values of n by Theorem VI. Then the values of c1 , c2 , ... , ck-t• and bi, b2 , ... , bk-p-t• are the same according to the enunciation in corresponding elements of rn 11 and Mn-k,k-t· In the former case we should have ck= n-2k+2-p,

bk-p = n-2k+2,

bk-p➔ t

= p,

bk-p+2=p-l,

... ,

bk-1 =I;

and in the latter case we have bk-p = p,

bk-p-1 = p-l,

... ,

bk-1 =I;

a.n-,,., we write Then

Thus

and Similarly, i = 2(a14f324-4a13a2 f31 f323+3a1 2a2 2f3 2f3 2)a"-4{3n-4 12 i; i:

= 2l;un-4) (n-4)-•(n-4) (n-4) +o(n-4) (n-4) 1\ A

µ-4

i\-1

µ-3

i\-2

µ-2

_4(n-4) (n-4) +(n-4) (n-4)} aAa :&.->--,,.-•~+,,.-•, i\-3 µ-1 i\-4 µ "' 1 I 469

185

A.

YOUNO

Then

½ni i>.,. = A4 (n-µ) 4 -4A3 µ 1 (n-Ah(n-µ)a+6A 2 µ 2 (n-AMn-µ) 2 -4A1 µ 3 (n-A)3 (n-µ)i+µ 4 (n-A) 4

(A =I=µ),

and 4. 'l'he equation (I) becomes, on putting in the values found,

3An 2 (r- l )(n-r-1)+ B(n-2)(n-3) r(n-r) = 0. We regard this as an equation for r. Unless it has a positive integral root less than n, f must be the n-th power of a linear form. When r is a root the other root is n-r; let r < n-r. Then, in any case, a0 = a1 =

... = a,_ 1 = 0,

whatever factor of/ is chosen for x 2• It may happen also that a,= 0, then a,+1 = a,+ 2 = ... = a,1_,_ 1 = 0, in which case / may be transformed into the form Otherwise every root off is repeated exactly r times and

I= if/'. Now, in this case, AJ2i+BH 2 is a covariant of ip of weight 4; it can therefore be written

Since this is zero, and if, has no repeated root, we must have B' = 0, and therefore (if,. if,) 4 = 0. Thus the solutions are (apart from a mere n-th power) J = x{x;-r; f =if,'; (if,, 11') 4 = 0. Ju particular, if, may be any cubic. 5. ()ovarian ts of weight 5 are included in the form

(AJ2i+BH 2 ,/)= (cf,,/). When x 2 is a factor of/, tbe coefficient of x~••-10 in (cf,,/) is 4B a/•, hence every root of/ is repeated unless B = 0.

Let

470

BINA.BT :,ORMS WITH A VANISHING COVARIANT

186

Taen (,/,,/) = f1:(: )ra,x~rx;-1 } {l:(4n-8-s) ,f,,~n-9-sx2•}

-f1:(:) (n-r)arx1-r-1x{} {l:s,f,,x1"-B-•~-t} = 1:(:) {r(4n-8)-sn}a,,f,,x~-,....._,x;+e-1 • Let us suppose that a0 = a 1 = ... = a,,_1 = 0, and consider the coefficient of ~n-6,-&x~P-6 in (,/,. /). It contains products a,,/,,, where r+s = 5p-4; when r > p, s < 4p-4, and,/,,, which is the sum of products of the a's of degree 4 and total weight s+4, must be zero. Thus the only non-zero product here is aP ,f,.,,_4 • Hence (;)

6

(A 0 iPP+BH!P) {p(4n-8)-(4p-4)n}a/ = 0.

There are three possibilities:

ap = O; Aipp+BH!P = O;

p

= ½n,

The last is a new case, and the conditions are satisfied for all values of A and B when every root off is repeated ½n times. It is obvious that this should be so, since f is the power of a quadratic and a quadratic has no covariant of odd weight. Let p< ½n be a solution of (I). Then every root off is repeated at least p times. If a root is repeated more often, it is repeated ½n, n- p, or n times. In the second and third cases, A/2 i + BH 2 = 0. In the first we must have one root repeated ½n times, and the others p times; and p must be a factor of ½n, Thus where ,f, has a non-repeated root. 'l'hen And, since ,f, has a non-repeated root, the equation

A' i;,.+B'(H;,) 2 = o must have r = l for a solution, and hence .B ' = 0.

Thus

{(,f,, ,f,)"', ,f,} = 0. This means that there is a functional relation between ,f, of order p and (y,, y,)4 of order 4p-8. Thia requires that, unless (y,, y,) 4 is zero, p =-= 8 or 4. The

471

BINARY FORMS WITH A VANISHING COVARIANT

18i

only case needing examination is the first. Here both (if,, if,)' and ip are octavics, there is a functional relation between them, and if, has a nonrepeated root. Then (if,, if,)'= >.if,.

(II) We write

if,= (b0 , b1,

••• ,

b8~x1 , x1 ) 8

and choose the variables so that b0 =b 2 =0,

b1 = I,

x 2 being an unrepeated factor of if,. Then, on equating coefficients in (II), we have b3 = 0, -24b4 = s>., b6 = 0, b8 = o,

Whence and

if,= (x14 + 10b4 x 1 xz3)(8x13 x 2 -10b 4 x 24 )

= F(F,

F)Z;

where Fis a quartic for which the invariant (F, F)' is zero. The meaning of this last case was pointed out to me by Mr. Grace some years ago. Thus when a covariant of weight 5 is zero, and no covariant of lower weight vanishes, the quantic is a power of a quartic or else of an octavic of the kind just described.

NOTE ON TRANSVECTANTS ALFRED YOUNG*

The calculation of transvectants of binary forms in terms of symbolical products frequently involves considerable arithmetical labour. The purpose of this note is to explain a method by which they can be written down at once. Consider the transvectantt

• Received 27 January, 1933; read 16 February, 1933. t Grace and Young, Algebra of invariants, 49.

472

188

NOTE ON TRANSYECTANTR

This may be written v! (m+n) v

(P) (azm v

bz",

m! n!

CzP) •= ~ (ac)A p!

(be)"

µ!

.:\!

ar;-ll

b:-"

c:-•

(m-.:\)! (n-µ)! (p-v)! ·

This is a particular case of the perfectly general formula

where the sum extends to all possible solutions in positive integers or zeros of the equations k

~

t=l

,\,+µ, = m,,

A

~ r=l

,\,+v,=n.,

~~A,.=m. r •

The proof follows the ordinary process. The result is assumed true when there are h different quantics on one side of the transvectant and k on the other. We then suppose that bt is really the product of two quantics, and so bn, 1, -- C;zPd;zq . Hence we operate on both sides with

a +c2 ob12 a ] 11' aJP = [ obu [ obi C

Ci

and after operation replace the symbol b1 by d. the left, On the right,

Then bt becomes, on

b••• IT 11 (a b )A , i •·•

-! I

Vt • r-1

}

I

"rl •

becomes, by using Leibniz's theorem,

where Pt+ P2 = V1, Prl + P,2 = A,1, Pt+~ Prl = p. The result then follows for h and k+ l quantics, and so the induotion is proved. 473

SOME GENERATING FUNCTIONS

By

ALFRED YOUNG

[Received and read 10 December, 1931.)

The primary object of the following is to obtain the generating function in two variables whose coefficients are the number of concomitants of degree 8, of the weights indicated by the indices of the variables, of a single ternary n-ic. The method used is an application of quantitative substitutional analysis as developed in my third paper on that subject•. It is proved that the generating function may be obtained from that for gradients by multiplication by factors of the form (1-y)(l-z)(1- ; ) .

The second part of the paper deals with generating functions of covariants of a system of binary forms of the same order; the generating function obtained is for covariants of the type T., .. ..... C,

where the permutations interchange quantics as such. As an immediate application the generating function for combinants is given. 'l'he third part of the paper extends the result for ternary forms to forms with any number of variables, and gives the generating function for seminvariants in terms of that for gradients.

I. Ternary forms. l. Consider a gradient X of a ternary quantic

n) A

l: (r,

8

xn-r-•xarxa•.

r,• 1

• Proc. London Matl&. Soc. (2), 28 (1928), 255-292.

474

426

A.

YOUNG

[Dec. IO,

We shall describe the coefficient Ar,, as having three weights n-r-8, r, 8, the first, second, and third. Thus the gradient has three weights, and the sum of the three weights is nS, where S is the degree of X. Gradients in which the weights are in descending order of magnitude will be considered. In the symbolical notation the quantic is written a:,;n, where

using dashes for convenience as the suffixes are used to distinguish different symbolical letters. The coefficient Ar,• is then an-r-•a'r a"•. The gradient X -- ITA"••• ,,, -- IT a,n-r,-s, a,,,,a,,,,, .

In order to apply the substitutional analysis we polarize this to make it linear in every· symbolical letter, and so replace a, by the n letters a11 a12 ... a111 , introducing a substitutional operator

because the letters are all equivalent. Thus where r is the product of the operators mentioned, it may be replaced by al;, The effect of a permutation (Pr) on a product Pr, P'r', or P"y" is to make no change, but it interchanges the pairs of products Pr', P' y; Q I I QII • QI II QII I ,.,,, ',., y,,., y ',., y. Thus the product above is unaltered by any permutation of the symmetric group of the w 1 undashed letters, or that of the w 2 letters with a single I the element is that just given. Then Next, in A1" we multiply the (s-1)-th column by z•- 3 and add to the a-th in regular sequence from the end, as far as a= 3; and thus obtain

Proceeding thus, we eventually obtain A"= A'"= j(l-z«,-r+a+l) ... (I-z•,-r+II) I, the last column being made up of units. tractions, we find that

Now, by means of row sub-

A'"= zn [ Il (I-z«,-«,+1-,)J A. r-,,,., ..,·,, T1r~ ... r,,,

where in F the number of sets of n equivalent letters which have r 1 in the first row, r 2 in the second, and so on is ..\,,r, ... ,h· Further, every seminvariant can be expressed linearly in terms of seminvariants thus defined. The weights of such a seminvariant are

A term of the concomitant led by this seminvariant has the same substitutional properties defined by the operator PN, but its weights are different from those of the seminvariant. Consider a term whose weights are m1 , m2 , ••• , m1,. Then w1 ~ m1 , w1 +w2 ~ m1 + m2 , and so on. Let 6 be the degree of the concomitant, the number of letters in PN or F is then n6. Let us use separate letters a, b, c, ... for the moment for each of these, aud consider them as ordinary symbolical letters so that

in the usual way. Then of these n6 letters, in any gradient of the term considered, m1 have the suffix I, and this gradient is unaffected by the permutations of all the letters with this suffix, i.e. it is unaffected by the permutations of a positive symmetric group of degree m1 • Similarly, it is unaffected by the permutations of a positive symmetric group of degree m2, and so on. Thus, calling the letters with symbolical suffix r 0r1 , 0r 2 , ••• , Or.,,,, the positive symmetric group of these letters Gr, and the product of these groups G, PN may be rewritten and supposed multiplied on the left by G. Then, as

490

442

A.

[Dec. 10.

YOUNG

in §5, GPN can be expressed in terms of forms P' uN derived from tableaux F' which are standard according to the sequence of 8 sets, and in which there is no overlappi,ng. Let us call the letters in the r-th row of F. which defined our concomitant,

Brl Br2 •• • Br•,• The concomitant may be developed into actual coefficients from its PN form, by first expanding according to the symbolical letters B, and then using polarizing operators. As an example, consider the seminvariant of the Hessian of the binary cubic, defined by A 0 A 2 or

(ai

Fa2aab1) - b2ba •

It is using B 1 for the first row letters and B 2 for the second. We polarize with

4/ 21 (a a~)a (b a~) (b a~) z, and obtain

!(ab }2a1 b1.

Consider any gradient Y of weights m1 , m2 ,

'1',,,.,,.,G~ y

••• ,

m11 •

Then

= L,\P' N',

where P' N' is defined by the sets (Jr- There are then just the same number of linearly independent forms P' N' as there are standard non-overlapping 8 tableaux. Next consider the correspondence between 8 and B; no matter how this is arranged, P' N' is unaffected by the permutations of {Bu B12 ... B1,,,}. Operation with this group places each of these letters in the first row of the tableau, completely filling it. Similarly the second row can be filled with Bn ... B;:..,· Thus for the terms of our concomitant each one of the kind P' N' can be expressed in terms of the same one. Thus the number of independent terms of weights mi, m2, ••• , m11 is the number of standard non-overlapping tableaux }". Let this number be

this is zero when m1 > w1 or m1+ m2 > w 1+w2,

491

1931.)

SOME GENERATING FUNCTIONS

443

the number of seminvariants of these weights. Then the number of gradients of weights m1, m1, ••• , mh is

Let 8,, be an operator which increases the r-th row of a tableau by unity and decreases the a-th row by unity.

'l'hcm

terms in

[wm wam ··· wh] is the number of m,, 1 1

1 •••

where overlapping terms are excluded and the indices Ar,, are such as to make the final rows w 1 , w2, •.. , wh. Thus, if D is the operator D

=

II (l-8r,,),

r-1 a>-..-1 · ·aA•) O(a1 2···, • s- -0·

There is a close relation between our seminvariant (a/a~• ... a~•) and the gradient where A,.,, is defined by

In this respect the operator O of equation (I) appears as the well-known algebraic operator 0 0 = ~~(n,.-a)Ar,Hl oA , , ,.,.

In the case of seminvariants of a single form the notation may be changed with advantage to (A >-•A>-, 0 1 ••• A>.") n

or simply where A,. is the number of sets of n letters which have r letters in the lower row of the tableau. Then (II)

O(A~ Ai' ... A!")= 0, 0

where O is the ordinary algebraic operator. 5. The seminvariants of a single binary form will now be considered. Those of the same weight will be arranged in a definite sequence. For this purpose two different laws will be considered, since each possesses a certain advantage; they will be quoted as sequence (A) and sequence (B); but, since the for;mer is that in general adopted, it will be assumed to be the one in use unless otherwise stated.

499

310

A.

[May 19,

YOUNG

(A) The sequence from the last letter: (A~0 A~• .. • A;•)

precedes

(A"O0 A"1 1 ••• A n11 •)

when the first ol the differences

which does not vanish is positive. (B) The sequence from the first letter requires that the first of the differences which does not vanish must be positive. A seminvariant which can be linearly expressed in terms of earlier seminvariants is called reductible, a seminvariant which cannot be so expressed is called irreductible. From equation (II) it follows at once that all seminvariants for which A1 > 0 are reductible according to either sequence. 6. Consider now seminvariant types of degree 3. They are given by the forms (a1°al'a3"). As in Q.S.A., II, § 28, we shall operate with the positive symmetric group of the n letters a 2 ; as in the place quoted this may be treated in two ways,. either as a sum of permutations of the n letters a 2 , in which case the effect. is merely multiplication by n!, or as bringing up all then letters a 2 into the first row and filling the A places of the second row by letters a 1 and a 3 from the first row in all possible ways. By equating the two results we get

(-Y

G) (a °a/a 1

3")

= ~ (:)

G=~) (ata2°a~-u+,.),

-i.e.

(-?(~) (;)(a1°al'a3")=I:(.\-;+µ) (:) (.\-=+µ) (a "a/a~-u+"), 1

which suggests a change of notation with the definition

{a/a 2"a{} =

(~) (;) (:) (a/a2"a3w).

Our equation then becomes (-)A{a1°al'a3"} =

500

~ (;) {a~+"-"a/a3"}.

1932.]

ON QUANTITATIVE SUBSTITUTIONAL ANALYSIS

311

More generally, if the orders nr are not necessarily equal, we obtain in the same way ( - )"2 {ai° a~• a~3 • • , aN = l: [,~3 ( ~ ) ] {ar-:>:11 a2°a;• ... a:,).

(III)

7. THEOREM II. of de,gree 3 :

The Jollowi,ng relation connects aeminvariant typu

where This is analogous to Stroh's equation*. To prove this we use the equations

oar}- (-)w-rl: {aw-ra 1 2a-

(W-8) r {a oa •aw-s}

12a,

to transform the last two sums, and we then consider the coefficient of

{ai°a2•a:-•}. When s ~ y 1 the last sum provides no part of this coefficient. A sum is required of the form

This may be regarded as a function of,\ of order f; let us replace,\ by -µ, then

(ft") is

(f-µ)(f-µ-1) ... (-µ+I)_ (-)l (µ-1)

e1

-

e ·

The function becomes

• Grace and Young, Algebra of invaria1it1, 64.

S01

312

A.

and the original sum is (IV)

(- )t

f {- )r (w-s)

Thus

,-o

r

[May 19,

YOUNG

(11-~- 1).

(r2+Ys-·r) = (- )Yi (w-s-ra-1)' Ya Y2

and the total coefficients of the terms a ::::;; y 1 are all zero. When a lies between y1 and y1+r2+1, then 0 ::::;;

w-s-y

3-

l

< y 2,

and hence the coefficient obtained from the second sum is zero, while no such terms are obtained from the other two sums. When 8 > y 1 +r2 it is necessary to consider the last sum; the coefficient of the a term is

f

,-o

{-)Y.+r( 8 ) (Ys+ri-r), w-r Y1

the coefficient of xY 1 in the expansion of (- )w+Y• [{l-{l+x)}•(I+x)Y•+Y1-w_ {l+x)Yt+Ya-w+a{l+x)Yt+Y•-w+1 - ... +{-)Y1+y,-w( and this is

8

w-r1-y3- l

{-)Y{ (Y2~Y1)-(~) (Y2~Y1~l) + .. . +{-)Y•

)

{l+x)-1],

(;J (~)]

={-)Y•+we-~:-1)•

In this case the coefficient provided by the second sum is the coefficient of xn in {- )w+y1 [1-{l+x)-l]w--•(l+x)Y•+'I'• = {-)w+Y•x'c--•(I +x)Y•+Y•+a-w, since the index of I +x is always positive. This gives {- )Y•+'l'•+l ( 8-yl

-1),

a-w+ra

and once more the total coefficient is zero. Thus the identity is proved. A transformation by means of equation (III) gives the following variation of the result just proved, simply and directly; we give it as a corollary because it is sometimes useful.

502

1932.]

0N QUANTITATIVE SUBSTITUTIONAL ANALYSIS

313

COROLLARY.

when y 1 +y2 +y3 = w-1. 8. It is to be observed that Theorem II may be applied to seminvariant types of any degree greater than 3. For by (III) we obtain in all cases (-)A 2 {a/a~2a~' ... a~•}= l:

({J{a~ +>-r a2°a3 a~ ... a~•}+K, 2

11

11

where K represents terms in which some of the indices later than that of a 3 are increased but none are diminished. Thus, in particular, we may suppose Theorem II always applicable to the first three quantics, and the result is true except for forms in which the weight is shifted to later quantics. The theorem may be applied exactly as Stroh's series or Jordan's lemma to prove that for degree 3 we may express a seminvariant type in terms of the types

where Further, when w= 3.\+I, ( alOal ar+l)

+ (ar+l a2Oal)+ (at" a~A+l a3°)

is expressible in terms of forms in which the highest index is increased. This is true also of the differences oa2Aa2A)-(a2Aa oa ") (a1 3 123> and so on. When we apply these results to the seminvariants of a single form, wo see that the lowest index is, of course, zero; the second must be even or by (III) the seminvariant is reductible. Let the second lowest index be 2.\ and the third .\3 • Then the form is reductible according to either sequence when .\3 < 4.\, or when .\3 = 4A+ I. We have seen ( § 2) that there is no relation between types whose tableaux have no overlapping, and in which the quantics are arranged according to the fixed sequence. There are then, in general, w+ I independent forms {a1°alaa"'}, and there are no relations other than those given by

503

314

A.

[May 19,

YOUNG

Theorem II. And thus for seminvariants of degree 3 there are, in general, no others than those which we have obtained. In other words, they are really irreductible. 9. The equation (I) for seminvariant types is one in terms of which all other equations between types may be expressed. This leads to a remarkable result when the equation is expressed in terms of the new forms introduced in § 6, i.e.

The equation (II) may now be written (V)

I:

(a,.a:,. a,.) {ai a~ 1

2 •••

aM=O;

the operator, of course, operates on the argument, and the numerical coefficients are brought outside. Thus the equation is

I:(.\,.+ I) {a1 1 a~2 ••• a~,+1 ... aM = 0. It is to be observed that these equations are entirely independent of the order n,. of the forms. The only occurrence of the order is when the index .\,. > n,., the coefficient ( ~,.) is zero, and the form {a1• a~• ... } is zero.

,.

Thus the relations between these types are independent of the order of the forms, so that we have · THEOREM III. By the introduction of a numerical factor the relations between seminvariant forms may be made independent of the order of the forms and the same as for perpetuants. In particular, seminvariants of a single form are irreductible or reductible independently of its order, according to any sequence.

IO. Consider the seminvariant, of a single quantic,

where ,.\ ~ ,.\_1~ '¼-2 ... ~ -'2·

If the order n of this quantic is less than .\,, this seminvari~nt vanishes owing to the factor (~) ; otherwise it is reductible or irreductible independently of the value of n.

504

Let us take n = .\,.

1932.]

ON QUANTITATIVE SUBSTITUTIONAL ANALYSIS

315

Then it may happen that

In this case, there is no such seminvariant for n = A,, so that the form is zero. This means that for any value of n this form can be expressed in terms of forms which have one index greater than ~- Hence the form is reductible according to sequence (A), but not necessarily according to sequence (B). It has been remarked that all types for which A1 is zero are linearly independent, and also that equations (III) enable us to express all other types in terms of these, then all our relations for reductibility may be derived from (III). In fact, they are the result of taking each letter in turn as the one with zero index and then using the fact that the letters are all equivalent. It follows from this that, if it has been proved that

is reductible, then also must

aAr aA•} {a 1OaA2 2 •·· r •• • & be reductible according to sequence (A), the indices being in ascending order of magnitude. For the reduction of the form of degree r might have been obtained from equations (III) [or indeed from equations (V)], and, in either case, it is appJicable to the form of degree 8, it being remembered that any increase in the later indices itself constitutes a reduction. We may state then 'l'uEoUEM

IV. A seminvariant, of a single quantic,

in which tlte indices are in ascending order of maqnitude, is reductible according to sequence (A) when where r ~8. It may be noticed that the consequence of putting r = 3 is the result already obtained, that for irreductibility A3 ;;;,: 2A2•

505

A.

316

[May 19,

YOUNG

11. The general identities proved in Q.S.A., II, 390-393, have already been referred to. They were proved for invariants, but the proof given is true also for seminvariants. The notation used differs from that used here in that the forms are defined by the numbers of letters of each set which lie in the upper row of the tableau instead of the lower row. To change the notation to that used here, each index ,\ must be changed ton-,\. Further, it is to be remembered that the result is given in terms of forms (a~ 1 a~2 ••• a~•), or rather of the forms (A~ ... A~"), and not of those denoted by {a~• ... a:•}. When the changes indicated are made we have an identity

where m-p = w-~')./3A"' to which the following meaning is to be attached. Each side is expanded, but only those terms which are of weight w, when x 0 , x 1 , ... , xn are regarded as the coefficients of a quantic, are retained. Then, if n8-w. Let m = w+p; then

n8-m = w+ll-p, and the result of equating the coefficients of m 10 gives

where R is the order of the seminvariant.

Also we obtain

(VII)

when p> R. 12. For an invariant this gives 0 A"°) (A" 0 A"• 1 · · A>.")-(-)W(A"••A~n-l ·110 1 ···11,

a relation obtainable at once directly from the consideration that in the tableau for an invariant there is the same number w of letters in each of the two rows, and these rows may be interchanged, the result being multiplied by (- )111 • It has been seen that, for an irreductible form, ,\1 is zero; it follows that, for an invariant, An-l is also zero. Also, if the form is irreductible according to sequence (A), the first of the differences

which does not vanish is positive; according to sequence (B) it will be negative. Further, when all these differences vanish the form is zero unless w is even.

507

A.

318 13.

THEOREM

V.

[May 19,

YOUNG

The seminvariant of order R, 2 3 (A"•A" O 2 AA3 · • • A"") n ,

ia always reductible, according to either sequence, when An-1> R .

From (VII) nR+l(AAoA>-2

""

0

2 •••

A"•-1-R-lA>-,.+R+l) = 0 n-1

n

•

and the result follows at once, for the form in question is the earliest term in the equation which appears. THEOREM

VI.

The seminvariant of order R, (A~• A~• .. . A!"),

is always reductible, according to sequence (A) , when the first of the differences

A0 -An-R, R-An-i• A2-An_ 2, .. • which does not vanish is positive ; and, if all these differences vanish, when w is odd.

This follows at once from equation (VI), 1••• A"") (A 0A·A"• 2 • · • A"·> n -- (-)W _!_ R! uf"\R(A""A""O 1 n •

More generally, from

.!_ f"\P(A"oA"• A"-,.-1-pAA,.+p) -(-)W l f"\R-P(AA,.+pAA•-J-p A"") O 2 ••• n-1 n (R-p) ! u O 1 ••• n •

pI u

we have 'l'HEOREM

VII.

The seminvariant of order R, (A"•A"• O 2 • · · A"") n ,

is always reductible, according to sequence (A), when the first of the differences Ao+An_1-An-R 1

R-2An-l•

A2-An_2,

which does not vanish is positive ; and also when w is odd and they all vanish.

14. In considering the reductibility of any form it will be usually written in the notation {a~1 a~' ... a;}, where here the indices are in

508

1932.]

319

ON QUANTITATIVE SUBSTITUTIONAL ANALYSIS

ascending order of magnitude and correspond to the suffixes in the notation used in the last few paragraphs. The irreductibility is ascertained first for the indices of lower magnitude. Thus, when .\2 is odd, there is no need to go further; .\1 must be zero. Then .\3 is equal to 2.\2, or greater than 2.\2 + l. For .\, we consider the form {a 1 a~• a~• a~•} as a seminvariant of degree 4 of the form of order n = .\,. The order of this seminvariant is R,, where

°

Then, writing we have

.\2

= 4.\+µ. .\, = 6.\+µ.+v,

= 2.\,

.\3

(µ.

= 0 or >

l),

R,=2v.

and

When .\ = 0, it is seen from Theorem VII that the form is reductible when v = l, for the differences are zero and w is odd. It will be seen later that, whatever .\ may be, the form is reductible when v = I ; or no ,covariants of degree 4 and order 2 can exist, a fact easily demonstrable -otherwise. 15. Another way of proceeding is as follows; .seminvariant type

operate on the

{a 1Oa2"a3"au,->--,,.} .t

with a positive symmetric group G which contains all the n letters a 3 , and n-p of the letters a 4 , the p letters omitted being in the second row of the tableau. To make matters clear we shall call Br the positive .symmetric group of the letters a,, and consider the equation derived from B 1 B 2 B 3 B, G {a/a/a 3,,.a4-~-,,.}

by using Gin the two different ways as in Q.S.A., II. This yields the equation

~

(VIII)

11

(p+u) {a Oa Aaw-p-u-Aap+u} p 1 2 3 C

=(-)w->--,~

When

p

(.\t'") {a1->--P-"a~+"aa°al}·

< .\, this gives a reduction for the left-hand side according to

509

320

A. YOUNG

[May 19~

sequence (B); that is we have a reduction for

I:(;) {a °aa"a;i->--sal"} 1

for the values 0, 1, 2, ... , A-1 of p. Hence we deduce a reduction for I;

rC~Xl) {ai°aa"a;i->--sa,s},

where x1 has any selected value. Thus any A consecutive terms may be expressed in terms of the rest and of earlier terms in the sequence (B). Let us choose them in the centre of the series; then, when µ,3 and µ,4 are the indices of a 3 and a4 , it is possible to express all the forms in terms of those for which the difference /J,3,.._,

µ,;::: "·

Thus all seminvariant types of degree 4 may be expressed in terms of those of the form where the identity of the letters is not fixed. Exactly the same process may be applied to seminvariant types of degree 3, where the quantics a 4_ 1 , a4 of highest index are selected in place of a 3 and a4 in the case of degree 4, and are treated in the sam& way. The result may be stated thus:

VIII. All seminvariant types of degree 3 can be expressecl. in terms of types of the form THEOREM

where

Thus, when using the sequence (B), it may always be asserted that the second index is equal to or less than w/

(!).

III. Relation of these forms to semin·varian~ expressed otherwise. 16. Consider the representation of a gradient of a single binary n-io in terms of the substitutional forms. The general gradient is

X

510

= A 0 Ai' ... A:,, 1

1932.]

ON QUANTITATIVE SUBSTITUTIONAL ANALYSIS

32I

the weight w will be supposed to be not greater than ½nS, where S is the degree. The factor .A;r will be replaced by nar symbolical letters, which appear for substitutional purposes as

A symbolical letter y appears usually in two ways, y 1 and y 2 ; for convenience they will here be written y and y'. A permutation (yS) will leave yS and y' S' unchanged, but it will interchange yS' and y' S. To express one factor .Ar we have a product

Then X, when written symbolically, is unchanged by any permutation of two of the undashed letters and also by any permutation of two of the dashed letters. We may therefore write it in the form

where Gna-w is the positive symmetric group of all the undashed letters, and Gw is the positive symmetric group of all the dashed letters. Using the identity where T .. , .., .....,. is the substitutional expression for the nS letters, we notice that TX=O, unless It

~

2, and a 1 ~ nS-w.

The case of chief interest is 7',,a-w,wX.

Here G,, 6 _w Gw is actually a l'r formed from a, tableau of this set; and the sequence of letters can always be chosen so that this Pr is standard. Since T=:i:P,N,M, and every P,N.111,Pr is zero unless r=s, and in this case (Q.S.A., III,. 264)

it follows that That is, in the expression of a gradient X in terms of the substitutional

511

322

A.

YOUNG

[May 19,

forms, the term arising from that T which corresponds to its weight is the -seminvariant (X) defined in our symbols by the gradient itself. The argument is stated for binary forms, but every word of it applies to forms with any number of variables; there is, consequently, no need to repeat it for higher forms, and, moreover, the expression would become too cumbersome. The only difference arises from the fact that for q-ary forms there are q-1 weights, and these in a "seminvariant" are equal to the number of letters in each row below the first. It is from thia point of view preferable to speak of q weights, and call nS-w the first weight in the binary case. We may then state · THEOREM IX. In the expression of a gradient X of a single quantic in any number of variables in terms of the substitutional forms, the weights of X being in descending order of magnitude, the term arising from that T which corresponds to the weights of X is the seminvariant (X) defined in .gubstitutional symbols by the gradient itself. The terms corresponding to .all later T's are zero .

17 .. It has been shown that, when X represents a gradient of weight not exceeding ½nS, the seminvariants (X) of a single binary form satisfy .a system of linear equations which may be expressed in the form (§ 4)

0. A r/Ar+l (X) = 0, it being understood that Ar+1 is a factor of X . Thus, in the same notation, (IX)

OQ(X)= 0.

Consider any seminvariant S = ~,\rXr; when we express it in terms of these substitutional forms, we have (X)

Elliott* has shown that all seminvariants can be obtained by a differential -0perator from gradients. He gives two equivalent forms for the seminvariant derived from the gradient X, one of which is

where 7/ is the order of the covariant, i.e. nS-2w in the notation here used. It follows at once from (IX) and (X) that this seminvariant is (X). • Algebra of quantics (1913), 224-225.

512

0~

1932.] THEOREM

form is

QUANTITATIVE SUBSTITUTIONAL ANALYSIS

323

Tlte seminvariant (X) of order R of a single binary

X.

18. The full expression for a gradient X in terms of the substitutional forms may be obtained directly by operation with the various T's. When we operate with Tns-s, ,.,., where ro is less than the weightw of the gradient, the result will be a sum of forms ( Y) which are seminvariant forms for that particular weight. The meaning, of course, is that here we are dealing not with the seminvariant or first term of the covariant, but with a later term, in fact, the (w-ro)-th term. When (Y) is regarded as a seminvariant, then in the expression for X it must be replaced by

The expression for X may also be obtained from Theorem X as follows. It will be convenient to use the notation (Qr X) for the sum of the seminvariants given by the various gradients obtained by the operation, each multipJied by the corresponding numerical coefficient. By 0-[(Qr X)] will be meant simply the result of operating with O• on this seminvariant. Then, by Theorem X,

and

We shall prove that

I + ... + (s-1)! (R+2s-2),_1 0,-l[(Q•-lX)]

= X-(-)S i (-t ,=,

When

8

= 1 this

I

(r-1) 8-l

r. (R+r+s)r

ornr X .

is Theorem X. Let us assume it to be true for any

513

A.

324

YOUNG

[May 19,

given 8 and use the value of (QB X) in terms of QB X given by (XI). introduce the additional term on the left

We

On the right the coefficient of becomes (

-

)"-•

(r-1)

l l ( )"+• s-1 s! (R+2s), (r-s)! (R+r+s+l)r-, - rl (R+r+s),

Thus our equation is true for s+l if it is true for s, and hence it is always true. Hence, increasings, we find THEOREM XI. The expression of a gradient X, for which n8-2w = R is not negative, in terms of substitutional forms or of terms of covariants is

X

= ,-o ~ S.I (R~2 S,) O•[(QB X)].

19. It is useful to remember that any seminvariant given by a substitutional tableau F may be expressed in the form (AB)w A~&-2w.

The letters A stand for all the letters in the upper row of F, and the letters B for those in the lower row. The actual expression of the seminvariant may then be obtained by repeated polarization with Aronhold operators, which replace the symbols A, B by symbols which directly represent the ground forms. For actual calculation this method is too cumbersome to attempt. Its importance lies in the fact that it is theoretically possible; and hence the use of single symbols A, Band the form of the seminvariant written above is justifiable. 20. Let (X) be any seminvariant of order R of one or more binary forms ; and let c~m be any other binary form. In the first place, l~t

514

1932.] m

=

0N QUANTITATIVE SUBSTITUTIONAL ANALYSIS

325

l, and consider the transvectant

Here (X) is used for the covariant instead of the seminvariant. As in §2,

(X)=GPNK,

where PN is given by a tableau F; this we shall write b1 b2 . .. bw.

Then, with the numerical coefficient given in Q.S.A., IV, § 1, we have

where

fw+R,R=

(2w+R) ! (R+I) (w+R+I)! w! ;

and

fw+R,u, I {a a }{b b c} - (2w+R)I w+I 1 · .. w+R 1 ··• '°

X (a1 b1 )

For

..•

(awbwHaw+l c) a10+2~ .. . aw+Ra

f w+R.w+l _ R f w+R.w (2w+R+I)! - (R+l)(w+I) (2w+R)I'

and 0 1 stands for the coefficient of the quantic Cz=(Oo, G1Xxv Xz>1•

The order of (XG1 ) is R-1, and the first transvectant of this with another linear form dz is

and hence the second transvectant of (X) with a quadratic c:e1 is

515

326

A.

[May 19,

YOUNG

Proceeding thus we see that the n-th transvectant of (X) with ex n is

Consider next the product (X) ex, or the seminvariant (X) 0 0 , where the order of the C quantic is one. This is a seminvariant of order R+ 1, and therefore expressible in terms of seminvariants T w+R+i,w· In Q.S.A., IV, Theorem III, the result of adding one more letter to a PN was obtained; since we are here concerned only with the terms of a particular T in that result (the other terms being necessarily zero), this gives us (using a notation which exhibits the tableau) {a 1 .•..•.. . . aw+R} b1 • . • bw

= [ 1+_1_ L (cb)] {a1 . ... . .. . . aw+RC}, R+2

b1 . •• bw

whence Then The O here contains the term

0 nCn-1 oC

n

and hence

n 1 (XC11 )D0 =(XCnDo)+ R-n+ 2 (XC 11_ 1 D 1 )+ R-n+ 2 (C,.D1 OX) . And hence ( {X),

c;+l) n = ( (X), ex••) ex 11

(R+2)... R+l (R-n+2)2 (XC,.)+ (R-n+2)2 (Cn+i OX). Proceeding thus we obtain THEOREM

transvectant

XII .

( (X), C:,;1•)"' =

If tlte covariant form (X) is of order R, tlten the

"i_"' (R+ 1) (n-µ) (R+n-µ+I),,_,._r (C,.+ror X). r=O

r

(R+n-2µ+1) 11_,.+1

The truth of this result has just been proved for n = µ., and for n = µ+ 1. We assume it to be true for n-1. By means of the process just

516

1932.]

327

0N QUANTITATIVE SUBSTITUTIONAL ANALYSIS

used one more symbolical c can be introduced, i.e. the ordf'lr of the 0 quantic can be raised from n-1 to n . Then the coefficient of (O,.+,.!l" X) on the right, is

(R+l) (n-µ.-1) r

(R+n-µ.),,_,._,._ 1 (R+n-2µ.)n-p

{

l+

µ+r } R+n-2µ.+ 1

+(R+l) (n-µ.-1) (R+n-µ) r-1 (R+n-2µ.)n_,.

11 _,,_,.

which reduces to the coefficient given. induction .

1

R+n-2µ.+l'

Thus Theorem XII is proved by

21. The last theorem may now be inverted with the result: XIII. If (X) is a covaria.n t form of order R, and O" represents a coefficient of a binary form of order n, then THEOREM

To prove this we may put in the value of ( (!l' X), 0:z:'' )"+' given by Theorem XII, and evaluate the coefficient of ( 0 ,.+. Qs X) in the sum on the right. This is

X

(R+ 2r+l) (n-µ-r)

= (n-µ.) :E (-)' (s) (R+2r+l) s

=

r

s-r

(R+n+r-µ.+1) 11 _,._• (R+n-2µ+1).n-l'-r+l

(R+r)!

(R+s+r+ 1) !

(n-µ,) (R+l)! s! :E (-)" (R+2s+1) [(R+r+l)+(R+r)] s (R+2s+l)! s-r R+l R+l '

In (IV), §7, these sums were evaluated. We write l: (-)"

(R+2s+l) (R+r+I) = l:(-)•-t (R+2s+1) (R+a+l-t) a-r

R+l

t

R+l

'

517

A.

328 and, putting TJ =

[May 19,

YOUNG

R+2s+ 1, g= s, ,\ = R+ 1, we obtain for the sum

Similarly

I:(-),_,(R+2s+1) (R+s-t)= -(2s-l). t R+l s-1 Thus the coefficient required is zero provided that s ~ 1 ; when s = 0, the coefficient is obviously unity. Thus the theorem is true. 22. Consider now a gradient X, of a single binary form /, for which n8-2w = R ~ 0. Let A,,. be the coefficient of greatest weight which it contains ; and let Then

"°

(X) =

"° (

X=YA,,..

(Y), / )"' +.\1 ( (OY), f

)"'+1 + ...,

where is not zero. Every term, except the first, can be expressed in forms (X'), where X' is a gradient which contains a factor Av, where v > p,. Thus (X) differs from a non-zero multiple of ( ( Y}, / )"' by earlier forms according to sequence (A). This may be generalized at once. Let

then X differs from a non-zero multiple of the continued transvectant

by earlier forms according to sequence (A). Thus the representation of covariants by continued transvectants of this nature is parallel to the representation by substitutional forms. A reduction in one case that is a representation in terms of earlier forms means also a reduction in the other case. It should be pointed out that the object of reduction here is exactly the reverse to that hitherto used in the discussion of transvectants. It has always been considered a reduction when a transvectant is expressed in terms of lower transvectants of forms obtained from the original forms by convolution; here it is a reduction when the transvectant is expressed in terms of higher transvectants of forms obtained from the original forms by devolution (the reverse of convolution).

518

1932.] TBBOREM

ON QUANTITATIVE SUUSTITUTIONAL ANALYSIS

XIV.

329

The covariant (XA,..) of a binary form differs from a

non-zero multiple of the tranavectant ( (X), I with forms obtaine.d from (X) by devolution.

r by higher tranavectanta off

IV. Covariant types of degree 4. 23. The covariant types will be written {a8 b"c"'d·},

where the indices are arranged in general in ascending order, and each indicates the number of letters of the corresponding quantio which lie in the second row of the tableau ; the form is that of § 6 with the appropriate numerical coefficients attached. Then, by §6, {a0 b" c" d•}- ( - Y· {b0 a" c" d•} is reductible according to either sequence. Thus for ,\ even the form belongs to {ab} and for,\ odd it belongs to {ab}' or {:}. Theorem II, § 7, provides a reduction according to either sequence when p. < 2.\. When µ, = 2.\, it gives a. reduction for

{a8b" c2"d•}-{b8 c" a2A d•}. Hence, when ,\ is even, {a8 b" c2" d•}-{abc} {a8 b"ciAd•}

is reductible, where {F} is the (PN) corresponding to the tableau F with the proper numerical coefficient. And, when ,\ is odd,

is reductible.

When

{a0 b>-c2>-+ld•}+{b8 c1 a2A+t d•}+{c0 a>-bl>-+ld•} is reductible.

Thus both

{ab..+,. reduction which may be written in the form (i) Let

{aob"c2>-+t d3'-H}-{c0d"a2>.+t b3"H} = o. And, since

520

">..' ( 1 and 7/ = 0 or 7/ > 1. Indeed, the whole result may be stated thus: THEOREM

XV.

The irreductible forms of degree 4 are included in X

= {a0b"c2"+tda>-+t+"},

521

332

A.

[May 19,

YOUNG

where the condition& are aa follows : T 4 X and T 1,X;

and ,\ even in the ftrat and

g=0 or g>l, 71=0 or 71>1,

odd,

T 1 , 2 X;

in the aecond caae.

f>0, 71=0 or 71>1.

T 8, 1 X and T 2, 1,X;

there is one aet of forms g> 0, 71 > 0 with ,\ ~ 0; and there is an additional set f = 0 or f > 1, 71 > 0, with .\ even in the ftrat and odd, in the aecond case. The generating function for T 4 X is

1 l+x6 l+x3 1 l-x12 1-x' l-x 2 = (l-x2 )(1-x3 )(1-x4 )' for T 8, 1 X, T 2, 2 X, T 2, 12X, T 1.X the generating functions are X

(l-x)(l-x2)(1-x4)'

x2 (l-x2) 2 (1-xs)'

(1-x)( l-x2)( 1-x')'

(l-x2 )(1-x3 )(1-x') · Multiplying these functions by the numbers

f,, la. 1,

/2, 2, /2. 1•, /1•,

and adding, we obtain (l-x)- 4 , the generating function for all forms of degree 4, and thus verify the accuracy of the work. V. 1.'he cubic and quartic. 26. This section is mainly inserted by way of illustration. For the cubic the forms (A~0 A~' A~~) have to be considered; the order of this covariant is

(It was shown in §5 that no gradients containing A 1 need be considered.)

By Theorem VII the form is reductible when

is positive, also when it is zero and the weight is odd. to be considered are, for w even,

• 522

(A:; Xi' x;•>, 0

Thus the only forms

1932.]

ON QUANTITATIV.h: SUllSTI1.'UTI0NAL ANALYSIS

333

,vhere X 8 = A 02 A 3 • Then (A 0 ) is the cubic, (A 0 A 2) the Hessian, (A 02A 3 ) the cubic covariant, (A 02A 32) the invariant. The products (A 0 )i,.o (X 1 )""' (X2)"'' are unrelated (a relation would give A 3 in terms of the other coefficients). 'rhis system is coextensive with (A~• xi• X~•), and hence is equivalent to it. The consideration that there are only two irreductible forms of weight 6 and degree 6, viz. (A 04 Aa2), (Al A 23 ), leads to the syzygy (Ao2 Aa)2- (Ao)2(Ao2 Aa2) = .\(Ao A2)a.

One is tempted to identify (XY) with (X)(Y), but, though there is a connection, this is not a simple one in general. The connection, however, becomes simple and direct when the substitutional forms are transformed from the natural to the seminormal or the orthogonal forms by the transformations introduced in Q.S.A., VI; this, however, must be reserved for a later communication, but we shall return to the subject here in § 28 and in Section XII. 27. The general form for the quartic is

(A~" A~• A~' A1•),

of which the order is R = 4.\0 -2.\3 -4.\4 •

By Theorem VII this is reductible when ,\0 < .\3 +.\4 , and also when = .\3 +.\4 and w is odd. In § 10 it was seen that this form is irreductible only when (A~" A~1 A~•) is irreductible for the cubic. Thus, for w even, the irreductible forms are

.\0

where X 1 , X 2 , X 3 have the same meanings as for the cubic and X 4 =A 0 A 2 A 4 ,

and, for w odd,

X 6 =A 0 A 4 ;

v"'X"•X"•X) (A 011-ox11-, l "'1.2 ~ 6 3 •

There is a relation between the gradients, for X 1 X 6 =A 0 X 4 •

523

334

A.

YOUNG

[May 19,

There arc only two irreductible forms of degree 4 and weight 6, viz. (A 0 2 A 2 A 4 ) and (A 0 2 Aa2) . There are also t\vo products of forms already obtained which cannot be equal, viz. (A 0 )(A 0 A 2 A 4 ) and (A 0 A 2 )(A 0 .A 4 ), so that (A 02 A 32) is reducible. We see that it will be reducible for any form of order greater than 3. The syzygy obtained for the cubic will also exist for forms of order greater than 3, but its expression will be modified by expressing (A 02 Aa2) in terms of products of irreducible forms . Thus the quartic forms will be

VI. Sequences. 28. In § 5 the seminvariant forms of a single quantic were arranged according to one or other of two sequences (A) and (B). These sequences were really sequences of gradients. Such an arrangement may be applied to the terms of a seminvariant when written out in full. According to a given sequence law, every seminvariant has a gradient which is its first term or leading gradient. Not every gradient could occupy the position -0f leading gradient; in fact, for any given weight and degree there must be the same number of possible leading gradients as there are linearly independent seminvariants. Thus a notation could be devised by which, for every possible leading gradient X, the seminvariant (X) is a definite seminvariant with X for leading gradient ; then all other seminvariants would be linear functions of these. The advantage of such a notation would be that a product (X)( Y) would differ from (X Y) by seminvariants having a later leading gradient. The main difficulty in adopting such a notation is that of determining what gradients can be leading gradients. In all cases so far examined (and a similar statement is true for ternary and quaternary forms) , the gradients X for which (X) is irreductible according to either sequence (A) or (B) are also leading gradients according to the same sequence. Although I am convinced that this is .always the case, I have not yet been able to prove it in general, and hence I can state it only as an empirical law. The consequences are important, for the form (X) could be defined as a form which has X for leading gradient ; with the resulting conclusions as to products. The role of this analysis would then be to determine which gradients can be leading gradients of a seminvariant. As has been already said in § 26, the need for this law will be removed when the serninormal forms are used. In the case of the cubic and quartic it is true not merely that there is a semin-

524

1932.)

ON QUANTITA'l'l\'E 8lf1:S'l'ITUT£0NAI, ANALYSIS

335

variant which has the leading gradient X when (X) is irreductiblc; but, further, when (X) is also irreductible its leading gradient is X. It follows at once that there can be no linear relation between the quartic seminvariants (A 0 )''"(X1 )"" 1 (X 2 )i.6 -2w~S-l, it is always possible to choose the numbers so that µ,6 > µ,. (s < S); and hence, when it is possible to find a symbolical product of degree 3-1 with X/AA, for leading gradient, it is possible to find a seminvariant with X as leading gradient, as has been proved already. When R < 3-1, there will be more than one of the

527

A.

338

[May 19,

YOUNG

numbers fLr equal to ~, for &

I:

r=l

fLr =

2w = 8~,-R.

It is sometimes quite easy to arrange that of these the cofactor of A,.,, i.e. the product obtained by removing all factors containing a,, has an earlier leading gradient than that for any other letter. It is convenient to arrange the numbers fL,, in a triangular table so that the value of fLra is on the r-th row of the a-th column, where r < s. The table fully represents the symbolical product. Thus, for example, the &

table for Il A 2,_2 = X is r=l

2

2

2

2

2

2

2

2

2 2

eontaining 2 in every position. The leading gradient is X, for the letters may be taken in any sequence with the same leading gradient always with a positive sign. And this is true whether the sequence is ~A) or (B). &

The table for Il A 2,-2+A, may be obtained by the superposition of the r=l

above on the table for TIA>.,• Hence, when the latter is a leading gradient, so also is the former. Other examples are 0

2

2

2

1

2

2

1

2 0

and

2

2

3

4

2

4

4

4

4

0

which give leading gradients

A 0 2A 3 A 6 A 6 and AoA2A,AuA12, the former according to either sequence, the latter according to sequence (A) only; both give irreductible forms (X) for sequence (A), the latter is reductible for sequence (B) by Theorem VIII.

528

H>32.)

ON QUANTITATIVE SUBSTITUTIONAL ANALYSIS

339

32. The sequences (A) and (B) are not the only possible sequences. In fact, the sequence underlying the whole classical treatment of invariant theory is actually the reciprocal of these sequences. The aim in the work of earlier writers has been to express the seminvariants virtually in terms of the leading gradient according to a sequence (C), which may be thus defined.

6

A gradient Il A,.,= X,., in which the suffixes a.re in ascendr=t

ing order of magnitude, precedes X,,. when the first of the differences

which does not vanish is negative. From the point of view of perpetuants, this sequence has the great advantage that the necessary and sufficient condition that X may be a leading gradient is that .\a-1 =

.\,.

This is in accordance with MacMahon's use of power ending product8*. The same thing may be seen symbolically. Every perpetuant can be expressed as a symbolical product in which some given letter appears in every determinant factor. When the greatest index of these S-1 different factors is less than twice that of the next greatest index, or when this greatest index is odd, the grade can be increased. After the grade is increased, one of the letters of the factor of greatest index may be chosen to be placed in every factor; and if necessary the process may be repeated. Thus we need only consider a product

where .\, is even and .\6

~

2-\,_1 ~ 2.\,_ 2 • • •

~

2.\2 •

'The leading gradient, according to sequence (C), is

where this is really the method of Gracet in dealing with perpetuants. It is easy to see that, when X is not a power ending product, then (X) is reductible according to sequence (C); for let X = IlA,.,, where .\, > .\,_1 ; • Elliott, ~lgebra of guantic,, 241, et seq • Algebm of i11vari1mts (1003), 326, et 1e2,

.t Grace and Young,

529

340

A.

[May 19,

YOUNG

then, by equation (I), § 4, O(AA, AA, ... AA.-1)

=

0

is an equation which reduces (X). And since the number of linearly· independent seminvariants is the same as the number of power enders, every form (X) is irreductible when Xis a power ender. Thus the law which is so far but empirical for sequences (A) and (B) is an established fact for (C). The weakness of sequence (C) lies in the fact that there is no clear indication of the order of the quantic necessary in order that (X) may exist and be independent of other forms (Y}, where X and Y are power enders. In fact, like the transvectant s(;)quence and the symbolical product sequence, the place of difficulty also is reversed in passing from sequence (C) to sequences (A) and (B). VII. Invariants. 33. The discussion of invariants is facilitated by the fact that the two rows in the tableau may be exchanged with an accompanying change of sign when the weight is odd. Thus (A 0 AA, ... AA)= (-)w(A 0AA,-A,_, AA,--A,_ 2 • • • AA,),

the order of the q uantic being here taken as the weight of the last letter. Hence for sequence (A) the form is reductible when the first of thedifferences .\s-.\s-1-,\2,

.\s-.\a-2-.\3,

which does not vanish is positive [negative for sequence (B)], and when all these differences are zero and w is odd. The form is also reductible when ,\. is I or ,\6 -1. The form is also reductible for sequence (A) [sequence (B)] when (A 0 AA, ... AA,.) is reductible for sequence (A) [sequence (B)], and when (A 0AA,-A,-, ... AA,-A,) is reductible for sequence (B) [sequence (A)] . These conditions are the complete conditions, at least so far as degree 6 is concerned, and every other form is irreductible. The invariants of degree 5 are then (AoA2,.. ...44,..H Ao,..+H~ A11),

where[, 'r/ are zero or greater than I, and 3n

We also have that is

530

=

24µ.+4f+2TJ .

n-6µ.-f-TJ

e~TJ-

~

2µ.,

rna2.]

341

0N ~UAN'l'i'l'A'l'!V.E 8l/HSTITlJTIONAL ANALYSIS

When g = 'Y/, we must also have w and therefore of forms

f even. '.fhis gives a. set

(AoA2µ. A4µ.+2E A6µ.Ht Asµ.Ht)·

Hereµ, and g may be any positive integers or zero. and the form is

Otherwise 'Y/ = !+3{,

l'his requires that 2µ, ~ {; also the form is reductible when { is odd. All possible systems of suffixes may be obtained by adding multiples of the four schemes 14=(0 0 2 4 4), 1 8 =(0

2

4

112=(0

2

4 12 12),

l1s = (0

2

7 18 18).

6

8),

2118= 2112+314.

Moreover,

That the corresponding products are leading gradients may be seen by the diagrams 0

l

l

2

2 2 4 4

l

2

1

2 4 4

l

l

4 4'

0

0

2

4

6

6

3

6

7

6 5 0

Moreover, these diagrams as well as the schemes are additive. Hence the former are irreductible and the gradients are leading gradients. The generating function is ( 1-xIO)( l-x20)( 1-xao) ·

The correetness of the result will again appear from the generating function . 34. The invariants of the quintic, by Hermite's law of reciprocity, These a.re

:should he related to those of degree 5.

>-,) (A >-,A>-,A>-·A· 0 2 :I 5 ,

where a.nd A5 ~A0 for sequence (A) .

531

A. YOUNG

342

[May 19,

Moreover, when A6 = Ao, we have A2 = A3 , and w must he even . Also this form is reductible when A2 +A 3 > Ao and when A2 +A3 = A0 , A3 odd. Thus, when (X) is an irreductible invariant, X is a product of the gradients

X, = Ai A 52, X 8 = A 03 A 2 A 3 Al, X 12 = A 04 A 22A 3 2 A 54 , Xie= Ao5 Al As6 ,

Xia= Aoe Al A5 7 •

Here

indicates the quintic syzygy; and also Xa2=X,X12 indicates a syzygy or a reduction. Consider the possible invariant forms of degree 16; they must be given by the gradients X 44 ,

X 4 2X 8 ,

Xa2=X4 X 12 ,

X 16 •

Also it is not difficult to obtain invariants with the above leading gradients, so that the forms (X 4 ), (X8 ) , (X12 ) are irreductible, and the forms (X4 ) 4 , (X 4 ) 2 (X 8 ) , (X8 ) 2, (X4 ) (X12 ), (X 16) exist. A relation between the first four would mean that (X4 ) was a factor of (X8 ) or of (X12 ), which could not be the case; hence (X 16) is reducible-and, indeed, (Xie)= (X,)(Xi2)-(Xa)2. For the leading gradient of the difference must be later than X 8 2 and hence cannot be either X 44 or X 4 2 X 8 The generating function for the quintic invariants is then ( 1-xio)( l-x20)( 1-xao), the same as for invariants of degrne 5. 35. Ju variants of degree 6 are given liy (Ao At,, A4,,+t Aa,,+H-•1 Aa,,+pA10,,+g),

where, since the order of the form given by the first five coefficients is positive, 3p ~ 4f+211, an-

Then

538

17" =

Z6

Yt-" Y2" = z3Yi+i Y2+1 ·

1932.]

ON QUANTITATIVE SU8S1.'ITUTIONAL ANALYSIS

349

This term in the final sum gives rise to six terms, one belonging to each function ; the corresponding values of p, K may be written p,, K,, . where r= I, 2, ... , 6: p1 =p, K 1 =K. These values are p, K : -p-K-3, p :

-p-K-3,

K,

and thus no confusion arises. 39. The same arguments may be applied when there are p sets of variables. We begin with the function (z, Y1, Y2, •· ·, Yv-1)

= ~ Z8 A8, ,,, r,, ... , r,,-1 Y1' Y2t ··•Y;J:-;_,

where A~. r,, rt, ... , rp- i is the number of types of degree 3 and of orders r 1 , r 2 , •• • , rv-t in the different kinds of variables. The concomitants are represented by tableaux of T"I, •t, ... , •p' where Then the variables y1 , y 2 ,

f 1 = yl'

f2

•••

are changed by the transformation

= Y1 1 Y~,

fa=

Y2 1 Y3,

•••,

g1' =

y;!_l'

or Y1

= {i,

Y2

= f1 f2 ,

. • •,

Yp-1

= f1 f2 • •• fp-1,

l

= f1 f2 · · • f P.

We write A

{

l:

l:

l: }' l:p+g-l l:p-2 l:p-3

uq= ~152 • ··5p

51

52

53

l:

.. . ~p-1 •

1 1 Then A S, r1, 12. • • . . , r>'-l is the coefficient of y'11+1 yr~ 2 + .• • y'~p-1 +1 in the expansion of

t:,.08+1

!:,./

And similarly the generating function is obtained for the case when the orders of the ground forms are not all the same. 40. The result of§ 37 may be used to obtain another form of generating function . We consider, first, types of even degree 2a of a binary quantic. Let A q,r be the number of types of order r, when the order of the ground form is q ; and let

539

A.

350

[May 19,

YOUNG

Then, by Theorem XVII, ycJ,(x, y)-y-lcJ,(x, y-1)

=

ao

I; (yll+1-y-q-1)28(y-y-1)-21+1z(l g=O

Now ycJ,(x, y) contains only positive powers of y, and y-1cJ,(x, y-1) only negative powers. Hence (XII)

(y-y-1)2&-lycJ,(x, y)

= 'i;.1 (-Y(28) y2(1-r)(l-xy2(1-r))-1+(-)'!(28) (1-x)-l+x(x, y). ,~o

r

8

Here x(x, y) is a function with the following properties: x(x, y)+x(x, y-1)

= o;

it only contains even powers of y; it cannot contain a term in which the index of y is less than -28+2. Hence

The coefficients A are functions of x. Since the expression on the left of (XII) has the factor y 2 - l repeated 28-1 times, each of the expressions obtained by differentiating the right-hand side of (XII) 0, 1, 2, .. . , 28-2 times with respect to y 2 and then putting y = 1 must vanish; we thus obtain 28-1 equations satisfied by A 1 , A 2, ••• , A,-1 • The method is too cumbersome for calculation, hut it yields at once the important fact that the denominator of c/,(x, y) is

11' (l-xy2r)(l-x)2&-2,

r=l

the numerator being a function tf,(x, y), integral and algebraic in x, y, which contains no power of:,; greater than 38-4, since the coefficient of x3'-3 is always zero.

540

1932.]

351

ON QUANTITATIVE SUBSTITUTIONAL ANALYSIS

41. When the degree 26+1 is odd the process is the same, but the form of the result is slightly different; for we then have yt/,(x, y)-1rl-•} 1 •

'l'he peculiarity of invariants lies in the fact that the two rows of the tableau may be interchanged; such interchange is accompanied with a change of sign when w, the number of letters in a row, is odd. Thus, for invariants,

543

A.

354

YOUNG

[May 19,

the numbers A2, • • • , A8 being in ascending order of magnitude. From our point of view the essential thing is the set of differences between the consecutive numbers 0, A2 , ••• , A8• We then write ir =

Ar+1-Ar,

and define the form in terms of these intervals,

It is convenient to use Sylvester's term excess of a gradient for 8A-2W, where S is its degree, Aits extent (i.e. the maximum weight of a coefficient contained), and wits weight. Then the excess of

{O A2

•••

.\s}

E 4 =8A4 -2w.

is

We shall require also the excess for part of the form, and so shall use Er for the excess of {O A2 ••• Ar} • Then it is not difficult to show that the excess in terms of the intervals is

Es= (S-2)(i,_1 -i1 )+(8-4)(i8_ 2 -i 2 )+ . ... The law of reductibility of types is taken here to be the same as for covariants for a single form according to sequence (A), Then a reduction is a relation wheres is a permutation of quantics, B ii:! numerical, and the first of the differences is_1 - j8_i, i 8_ 2 -js_2 , ••• which docs not vanish is positive. 46. Consider then where Then there is always a reduction when Er (r < 8) is negative.

½E4 =i3 -i=7J~O, For invariants,

{XIII)

544

E 3 =i2 -i =

f ~O.

Thus

!932.]

0N tiUAN'.rI'l'ATIVB SUBSTI'fUTIONAL ANALYSIS

355

Thus the form is reductible when the first of the differences i,-i, i 3 - i2 which does not vanish is positive. Also, when the dexter is reductible according to sequence (B), the sinister is reductible sequence (A). Hence, by Theorem VIII, there is a. reduction when any i,. is less than i,. This does not, however, yield any fact not obtained otherwise for degree 5, but it is useful for higher degrees. When there is a relation of the form (XIV) where R is a linear function of earlier forms, either (i) according to sequence (A) or (ii) according to sequence (B), then a similar relation is satisfied by (XV) the sign of every transposition in s being altered when µ, is odd. This follows at once from the equations of §IV, for 8 < 5; that it is true for invariants of degree 5 follows from equation (XIII), which is a.11 that is required here. I am convinced that the above statement is true for all values of 8, but cannot give a formal proof. When the intervals given refer to the leading gradient instead of to the substitutional form, its truth is clear. For, if we take K to represent the symbolical product form of (XIV), then that of (XV) may be taken as K. Il (a,.a,) 1'-,

,vhether the sequence is (A) or (B). We need then consider only the cases i, = 0,

i3 = 4i+f,

i2 = i+f,

that is the invariant C

When i

= [i i+f

4i+t

0] .

> 0, g > 1, the only relations of reduction for [l-(-)i(a1 a 2 )] 0

= R,

[l-(a,a6 )] 0

O are (see § IV)

= R.

That is, among the 5! functions obtained from O by permutation of quantics thirty are independent. Taking the permutations according to this analysis, we may represent the functions as (i even)

1. T 6 +2. Tu+2. T 3, 2 +1 . T3,1:+l. T 2,, 1 ,

(i odd)

1. T,, 1 +1. T3,2+2. T 3, 1,+L T 2,, 1 +1. T 2, 11 .

545

356

A. YoirNu

[May rn,.

It will be remembered that H'., ... stands for Af., ... forms belonging to this T*. The form is {0 i 2i+t 6i+2g 6i+2t}, and the diagram i

i+t

2i+t 2i

i

2i+t 2i+t 2i+t

2·i

0

shows that there exists an invariant, with a corresponding leading gradient, which exactly fuJfils the conditions. When i > 0, f = 0: i even, C belongs to {a1 a 2 aa} and to {a 4 a 6}; substitutionally it contains

1 . T 6 +1. T,, 1 +1. T 3 , 2 • i odd, G belongs to {a 1 a 2 aa}' and to {a 4 a 5}; substitutionally it contains

1 . Ta,12+1 . 1'2, 1 i . 'l'he same diagra.m may be used as before to establish the existence of the invariant. When i > 0, g = I, G obeys the same relations as

and the forms are

1 . T,, 1 +1. Ta. 2 -l- I . Ta, 1,+ 1 . 1'2 ,, 1 • When i

= 0, g >

I.

the forms are w even,

I . T 6 -j-l . T,, 1 +1. Ta, 2 +1. T 2, , 1 ;

and w odd, Finally, when i = 0,

1. T,, 1 + l . Ta, 2 -j- l. T 3, 1'·

f=

l, there is one form

(0

0

1

2

2)

• See Q.S.A., III, § 6.

546

1932.]

ON QUANTITATIVE SUBSTITUTIONAL ANALYSIS

367

of the nature T3.1,; for this we may take the diagram 0

1

1

0

0

I

I

0

I 0.

4 7. The results now may easily be verified with the generating functions. The invariant is of the form

[i+µ, i+,-i+[,

4i+,-i+[,

,-i],

or, in the other notation,

{O i+,-i 2i+2,-i+{ 6i+3µ.+2{

6i+4,-i+2{}.

The generating function is ~Arxr, where r = 6i+4,-i+2{; following Elliott (loc. cit., l 38), we call r the extent of the invariant. Its form corresponds to the common denominator of the functions

D = (l-x4 )(I-x6 )(J-x8 ); the indices corresponding to the coefficients of µ. and f are double their coefficients, owing to the change of sign of transpositions when I-' is odd, and the peculiarity of the case g = 1. When the forms corresponding to the va1'ious T's in the last paragraph are picked out, the following generating functions are obtained; the numerator alone is given in each case, the common n-2k or l>y. As before, we have to consider a matrix D belonging to T n-k, k and a matrix fl corresponding to a distortion of y places to the left in

T n-k-l, k+l• 581

464

A.

YOUNG

(Jan. 19,

The equations for A, A' are simplified by writing

Then, from the equations of § 10, it follows that, when a,, a,+1 lie in the same row of F/, but their interchange changes Fr into F,,

When (a,aH1) changes F,' into F,,' but leaves Fr unchanged,

,vhen a,, Ui+1 lie in the same row of F,' and the same column of Fr,

The tableaux will be defined by their second rows :

Fr: b11,b11,···b11k;

Y1 -,.]

=

[PA,.flh>-],

where tlte rozcs and columns are defined by the tableaux F>-, F,., icith second row suffixes and u·here

,.. PM= [F>-h= fl (A,-2r+I), r=l

1-

J)>.,.

= IT B~", r=l

The second result may be verified by the same method as the proof of Theorem X ginn here. It gives the expression for the permutation

which inverts the sequence of letters and may be called the inverting permutation; the result is as follows . THEORE)I

XII.

The exact seminormal matrix

589

,..

4 '"')

A.

[Jan. 19,

YotrNG

is tlte expression for tlie inverting permutation in the representation T n-A·, k• -u·liere I,

mx11 = II

f'=l

B:

B:

11 ,

B7;" = 0, Yr< n+ 1-xk-r+l•

= (xk-r+l-s-2k+2r-l +s) (n-2r+2-s-xk-r+1-,),

11

Yr= n + 1-a·k-i-l-r-s,

B;V = -s(n-2k+s+ 1), xk-r+l-s < n+ 1-y, < xk-r+!?-•· V. Algebraic equivalents of seminormal units. 16. The relations between the seminormal units and the natural units

are, of course, inherent in the equation jJfDM- 1 = G

connecting the seminormal matrix D to the natural matrix C. In a few cases the relations between the units may be very simply expressed. THEOREM

XIII.

The seminormal units mu, m11, m11, m11 are gfren by

mu= f/(n ! g)·P1 N 1 P 1 , m11=f/(n! y)N1 P 1 N 1 , m11 =p 11 =J/n! P 1 u11 N 1 , m11 = 1/gyN1 u11 P 1,

where g is the product of the orders of the groups of P, y is that of tlie orders of the groups of N and p 11 is the natural unit.

In every case the matrices M, M-1 have the property that their elements

mr,, m;, are zero for r > s, and unity for r = s.

To find the seminormal unit m11 in terms of the natural units, we put for D that matrix which has du= 1, and all other elements zero. Then

Hence

S90

1933.]

Ox

QL-\.XTITATIYE St;BSTITt'TIONAL AXALYSIS

4i3

Xoweverypermutation of P 1, expressed as a seminormal matrix, has zero for every element of the first row and column, except for that on the leading diagonal which is unity. Hence IDn P1 = Yr»n;

this establishes the value of m11 • The value of m11 is obtained in the same way; that of m11 comes even more directly from the matrix equation. Finally, N 1 m11 P 1 = gym11 • mil= W 1 all P 1.

Hence

The value of,\ is obtained at once from the product ro,1m11= roll.

The exact units are the same for ro 11 and ro11 , hut for ro 11 we must multiply by [F1 ] 1 /[F 1]i and for m11 by [Fi] 1 /[F1 11 ; see Q.S.A., VI. §18. COROLLARY.

It follows from the above reasoning that

TP 1 =gm 11 ,

TN 1 = ym11 .

17. A representation Ta,a, ... aA is defined by the lengths of the rows of its tableaux, it might equally well be defined by the lengths of the columns /31 = h, /32 , ••• , /3k· Let us suppose the letters a 1 , a 2 , ••• , an which are the subject of permutations to be really h-ary Yariables. Then with any column b1 b2 • •• bp in the tableau may be associated a determinant

and with the whole tableau the product of these determinaat factors. THEOREll

XIV. X

The product

= (a 1 a 2 ••• ap,)(afl,+1 ap,+ 2 ••• afl,+fl,) •••

llas the substitutional propertie,s of m11 and no others.

It is seen at once that Hence T' X is zero when the first of the differences l-'1-,.,1, a a ' 1-'2-l-'2, a a ' which does not vanish is positive. Also when /31' > /31 ,

•••

N'X=O,

591

4i4

A. Yol;:~w

[Jan. 19.

for the variables contain no more than /31 possible second suffixes. Similarly, this equation is true when the first of the differences

which does not vanish is positive. Thus it is true unless T' = T. Hence T'X=O, T'=T.

unlesiHence which proves the statement. COROLLARY .

y

The form, in single non-homoaeneous i-ariable.s,

= fl.1, 2 .. .. ,/l, fl./l,+1,/l,+2, ... , /l1+/l: ••• fl./l,+/l,+ ... +flk-J+l, ..• , /l1+/l,+ ... +~••

u.:liere

A -{a1 a 2 • •• a·13,1Vall,-lall,-2 ul,2, ... ,/l,l 2 • •• a/l,-1•

has the same property.

18. By operating with different permutations on the function X of the last paragraph, all kinds of products like X may be obtained; their substitutional form is given by the operations of these permutations on ro11 . Thus there may be obtained a linear function X, of such products corresponding to every unit m,1 ; this will be called the symbolical product equivalent of the tableau F,. In exactly the same way we may obtain from r a determinantal equivalent of the tableau F •. The symbolical products may be multiplied out, in which case X , n

becomes a sum of products TT ar, u with numerical coefficients. These r=l

terms may be arranged in a sequence just as the sequence of tableaux, the second suffix ur defining the row of the letter a,. XV. The first term of the symbolical product equfralent of tableau F, according to any principal sequence is that defined by the trtlJ/erm itself. THEOREM

a111J

Here a principal sequence is one defined as such in § 2. The term defined by the tableau itself is that obtained by writing for the second r-uffix ur the number of the row in which the.letter a, lies in the tableau. 592

1933.]

Ox

(lL.\XTITATIYE Sl.BSTITt:-TIOXAL AXALYSIS

4i5

The theorem is ob,·iously true for the tableau F 1 . \\'e assume its truth for all tableaux FT when T ~ u. Then, when u u. Then where pis the inverse of the axial projection of a,a,+ 1 in either tableau. The earliest term of X, is S,, that given by the tableau F,. Hence the term S,, given by the tableau F,,, which is obtained h~· the transposition (a,a,+1) from ST, is certainly a term of (a,.a,.+1) X" while it is not a term of XT, hence it is a term of Xrr . ~Ioreover, if possible, let S be an earlier term of X,, than that defined by F,,. Then (a,a,+1) S = S' is a term of FT. By hypothesis, S' is a later term than S according to any principal sequence. Taking the sequence from the first letter, if S' differs from S, in the first r-1 letters, then S' (and also S) is later than S, and also S,, in this sequence. In the same way, taking the sequence from the last letter, if S' differs from S by a permutation affecting one of the last n-r-1 letters, it is later than ST in this sequence, and hence also S is later than S,, in this sequence. Thus S cannot exist. And, in all cases, the term defined by the tableau F. is the earliest term of the function X,. COROLLARY. The Junction Y. defined by the tableau l\ in the last paragraph ha8 for its first term the term defined by the tableau.

The above reasoning applies to this case. VI. The regular group matrix. 19. The regular group matrix R of the symmetric group of n letters containsn! ro,vsandcolumns. It is usually defined as the matrix [xrQ-1]. where Pis the same for each row, Q for each column . and P and Q are respectively each operation of the group in turn; :i:J> then represents a set of n ! arbitrary variables. Then there ·is a non-singular matrix A for which A- 1 RA=D, where D is a matrix which consists of "£.f smaller matrices about the principal diagonal and zeros elsewhere. These smaller matrices may be made to consist of a succession of sets each set containing f identical group matrices of order J. Further, we may suppose A chosen so that these group matrices are the exact seminormal group matrices. The matrix A 593

A. YocsG

476

[Jan. 19,

consists of a series of bands of columns. Corresponding to a particular group matrix of D of order f, there is a band off columns which occupy the same position in A as the columns on which this matrix lies in D . And further, corresponding to the f identical group matrices in D, there will be f bands, off columns each, which, however, are not identical since A is non-singular. In the same way there are bands of rows in A-1 • Schur has shown that the only matrices permutable with D are of the form K, where in K the sub-matrix defined by the J2 rows and columns, occupied by a set off identical group matrices of order fin D, is

where Brs are f 2 arbitrary constants, and all the other elements are zero. Then A may be replaced by AK, as the most general form of A. This, in effect, replaces the f bands correspoi:iding to a particular representation by any f linear functions of them which are independent. Let us now consider a transposition ro = (a .. a .. +1) of a pair of consecutive letters in R; the coefficient of x...,. is represented by ½n ! quadratic matrices about the leading diagonal of the form

In D we must consider each group matrix separately, and the coefficient of x...,. is either + I on the leading diagonal or else a quadratic matrix of the form

[-p 1-p] I+p p

about this diagonal, and zeros elsewhere. Thus the equation

RA=AD giYes us a series of equations, of one of the two forms (i)

[ o1

1] [""er] A,er = ± ["•er] A,er '

0

whence (ii)

594

I] ["r" ",.,] [.\.er "'T] [[O 1 0 A,er A = A A,T +P 1,

14

1

p

1-p]

p

'

0~

1933.]

QUAXTIT..\TffE SCBSTITt:TIOX..\L AX..\LYSlS

whence

(1-p}A,.- = -pA,,+Ar,,},

A,.. = -pA,.. +(l+p)A,.,

(1-p}Ar.-= -pA,,+A,,.

A,.-= -pA,.-+(l+p}A,,.

Similarly, if

l

j.

A-1 = [µ,,],

(l+p)µ .., = -pµ,,,.+µ,,, (l+p) µ.., = -pµ,,+µ,,. Nowtherowsandcolumnsof R maybe supposed to be defined by , which is nothing more than then letters a 1 , a 2 , ••• , an arranged in some particular sequence. Then, in the abow, if r is the numerical equivalent of , sis that of (a.,a.,+l). Consider the last column of a band in A, the remaining /-1 columns of this band are obtained from it by the equations (ii) above. The elements of this last column will be written A,1 , where r has the values 1, 2, ... , n!, corresponding to then! sequences . Corresponding to r = 1, the letters in are arranged in order of the suffixes. An arbitrary function of n variables has, in general, n ! different values corresponding to the possible n ! permutations of the variables. We may then suppose A,1 to be such a function h).

For orthogonal transformations the relations between different columns. are given by

v'(l-p2),\~ =

whence

-p~+,\,,.

The remaining I'-1 columns may be calculated by means of this equation from that already determined. 23. As an example, we consider the double regular group matrix of degree 3. The sequence of arrangements is taken to be abc, 1

bac,

acb,

bca,

cab,

cba

2

3

4

5

6

The suffix r of each variable represents the permutation which changes. abc into the corresponding arrangement r. Then X1+Y1 X2+Y2 Xa+Ya Xs+Ys X4+y4 x6+Y6 x2+Y2 X1+Y1 X4+Ys x6+Ya xa+Y6 xs+Y-1 R1=

xa+Ya Xs+Y4 X1+Y1 X2+Y6 x6+Y2 X4+Ys X4+Y-1 x6+Ya X2+Y6 X1+Y1 xs+Ys Xa+Y2 xs+Ys Xa+Y6 x6+Y2 X-1+Y-1 X1+Y1 X2+Ya x6+Y6 X4+Ys Xs+Y-1 xa+Y2 x2+Ya X1+Y1

1

A1 = y(l2)

602

v'2

2

0

0

2

\,12

y'2

2

0

0

-2

- y'-l

y'2

-1

,13

y'3

1

-,12

y'2

-1

-v'3

y'3

-1

y'2

v'2

-1

y'3

-y'3

-1

y'2

y'2

-1

-,13

-v'3

1

-v'2

.

1933.]

ON QUANTITATIVE SUBSTITUTIONAL A~ALYSIS

485

'The orthogonal double group matrix T 21 is then

where

Xu+Yu

X12

Yu

0

X21

X22+Y11

0

Y12

Y21

0

Xu+Y22

X12

0

Y21

X21

X22+Y22

Xu= X1+x2-Hxa+x4+x:.+x6), .X22 = x 1-x2-½(x4+x5-x3-x6);

and Y 11, Y 22 are the same functions of y :

Xu=½ y3(x.-xs+xa-X6), X21 = !- v3(x:;-X4 +xa-X6), Y12 =

½y3(ys-Y«+Ya-Y6), Yu=½ y3(y4-Ys+Ya-Y6).

It is necessarily irreducible. 24. Consider next the representation Tw

Here

>i11=(A1-A2HAa-A«) = (A1'-A2'HAa'-A4'}.

There are four values of r for which >..;1 , and four for which ->..~,. is (A 1' -A 3')(A 2' -A 4').

In the first four cases, >..,.1 is (A 1 -A 3)(A 2-A 4 ), in the other four >..rf is (A 1-A 4 )(A 2 -A 3 ); hence

(A 1-A 3 )(A 2-A 4 )+(A 1-A 4 )(A 2 -A 3 )

= 0.

That is, if A 1, A 2, A 3, A 4 represent points on a straight line, the pairs A 1 A 2 and A 3 A 4 must be harmonically conjugate. This will, of course, give virtually the same double group matrix as in the former case; there are, indeed, four times as many variables, but they correspond in sets of four to each of the former variables. Similarly, for the representation T 33 (which again is practically the same as T 32 ), the pairs A 1 A 2, A 3 A 4 , A 6 A 8 , on which >..,.1 depends, must represent pairs of points mutually harmonic two and two. Such pairs are known to exist and hence the double group matrix can be transformed into the required form. But for a representation Tkk• where k > 3, since it is not possible to find more than three pairs of mutually harmonic points, it is not possible to express the group matrix in the form suggested. It is not difficult to show that, apart from the representations of unit rank, the only representations for which this is possible are T fl-I, 2 , T 33, 603

486

A.

YOUNG

[Jan. 19,

and the corresponding representations when rows and columns are interchanged in the tableaux, i.e. T 2, 1i and T 2,.. VIII. Concomitants given by seminormal units. 25. In what follows the results which have been obtained here are applied to the theory of algebraic invariants. The general method of the applications of this analysis to invariants has been explained (Q.S.A., VII). In the first place, with the use of natural units n,,, concomitant types take the form where G is a product of positive symmetric groups, expressing the facts that the first n 1 letters are equivalent as all representing the first quantic, the next n 2 letters are equivalent and so on. Here we are dealing with the seminormal units mr, and concomitant types of the form

Gmr,X. In both cases the second suffix 8 of the unit may be always taken as the same; in such applications we are only concerned with one column of the matrix. Now the operator G really means taking a sum of different terms, or else a number of elements in the column. These are obtained, from the one element expressed, by permuting the letters belonging to each quantic among themselves in all possible ways; the actual numerical coefficients attached to the different elements and units are given by Theorem V; and we see that in Fr no two elements belonging to any quantic may lie in the same column. It is not contemplated to write out the expression in full, but the power to do so gives complete definiteness to the abbreviated expression.

26. A concomitant of a single quantic requires another substitutional operator r in either set of units. This represents the fact that all the quantics concerned are the same, or that each set of n letters is interchangeable with any other set. Thus the full expression is rGnr,X, or rGmr,X. Like G, the operator r represents a sum. Any one of the units occurring in this sum might be taken for nr, (or mr,) to give the expression, but for the sake of the developments of the theory the earliest unit according to the sequence of tableaux selected is chosen. This unit or its tableau defines also a gradient, a product of the coefficients of the quantic; and, conversely, the gradient defines the concomitant. Every gradient defines a concomitant in this way, some of these concomitants are zero, and some can be equally well expressed by earlier gradients. A 604

1933.]

0:s

QUANTITATIVE SUBSTITUTIONAL A~ALYSIS

487

concomitant as thus defined by the earliest possible gradient is called irreductible ; naturally the term implies also a definite sequence. There is a distinction to be remembered between the n.a tural and seminormal units; in the natural units the letters of permutation are all on the same footing and their sequence is of no particular consequence until the sequence of tableaux has to be determined; in the seminormal units the sequence of the letters is of the fundamental texture of the theory. The terms of the sum given by r in this latter case are to be supposed calculated as from the values of the transpositions of consecutive letters, they are not arrived at by mere permutation. Further, in the natural units any pre-arranged sequence of tableaux might be used, and thus any sequence of gradients might be conceived as the foundation for the representation of concomitants. In seminormal units a practical limitation in this matter is laid upon us by the sequence used in the transformation from natural to seminormal units. There is still a considerable choice, as is shown by Theorem IV, but in seminormal units the sequence of tableaux must be a principal sequence, such as is defined in§ 2. 2i . Consider the transformation from natural to seminormal units. The natural matrix C becomes the seminormal matrix D, where

D

= .M-1 CJJ,

and ..H, ,.lf-1 are matrices with nothing but zero elements below leading diagonals of positive units. Thus the natural unit n,, becomes

where 1n, rn' are elements of .Zif, .JJ-1, respectively, and ro ... v are the seminormal units. The coefficient of ro,.v is zero unless u ~ r, v ~ s; also the coefficient of ro,. is unity. Hence an algebraic function given in natural units in the form

becomes where Ar= I , when we change to seminormal units. As has been observed, we are only really concerned with one column of the matrix; the transformation leads to several columns of the seminormal matrix, but they are all proportional to each other, and they may be absorbed into one by the change from X to X ' ; thus the 605

488

A. YouNo

eoefficient ~u only really relates to the first suffix of the unit. at once from the above :

[Jan. 19, We deduce

THEOREM XVIII. The difference between the algebraic forms or concomitants given by the same given tableau by natural and by seminormal units is a linear function offorms given by earlier tableaux in any principal .sequence in either set of units. The result as to concomitants follows at once when nr, is taken as the earliest unit in the concomitant ronr.X. 28. It is necessary to be clear as to what constitutes a principal sequence. This last theorem shows that, in the case of concomitants of a single form, the sequence, and consequently the determining tableau Fr, may be selected for the natural units first, and then the sequence of letters of permutation determined accordingly as a necessary preliminary to the transformation to seminormal units. Thu~, for binary forms under certain eonditions, a gradient

A~• A~• ... A~• has been found to represent an irreductible seminvariant. This is first polarized o-1 times in order to obtain o different sets of letters; and a particular term in. the result is selected. In the sequence of letters of permutation the letters corresponding to a coefficient suffix zero are taken first, then those of another, and so on, until these are exhausted, then those of suffix 2, and so on. With .such a method both the sequences A and B are principal sequences. Quite another sequence is used for the invariants of the quaternary cubic (Q.S.A., VII, section XI), but it-is not difficult to see that the sequence of the letters of permutation may be selected, so that this also is a principal sequence. The sequences O and D would require the separation of the letters of the different sets; the sequence of letters of permutation being begun with one letter from each set, then a second, and so on. They do not appear to be suitable sequences for use with seminormal units. 29. An empirical theorem has been stated (Q.S.A., VII, section XII, § 51 ). No proof had been found, but very strong experimental evidence of its truth had been obtained. This theorem will now be demonstrated. With the demonstration its scope is greatly extended, and accurately defined. It may be regarded as the fundamental theorem of this approach to the 606

1933.]

ON

QUANTITATIVE SUBSTITUTIO:SAL ANALYSIS

489

algebra of invariants. It may be enunciated thus: XIX. With any principal sequence, the leading gradient of an irreductible concomitant git:en in seminormal units is that which defines the concomitant. And the same gradients define the irriductible concomitants in natural units. THEOREM

The last part of the enunciation is an immediate consequence of Theorem XVIII. The first part follows from Theorem XV, § 18, when it is remembered that the tableau taken as representative of the whole form is that of the earliest unit in the sum. The theorem quoted applies to every term of the sum, and hence the leading gradient is the leading gradient of the first term. The second empirical theorem of the last paper follows at once from this, as was pointed out there. It thus becomes a demonstrated theorem and is no longer empirical. It is of much less importance. 30. The product of two concomitants has for its leading gradient the product of their leading gradients. And thus it follows that a concomitant given in seminormal units by the product of two leading gradients differs from the product of the corresponding concomitants by concomitants which are given by later gradients. This fact is given by Theorem IX, independently of the theorem just proved. In looking for the complete irreducible system of concomitants of a quantic, all leading gradients have to be considered; but those which are products of other leading gradients may be rejected at once as reducible. In binary forms it is often easy to show that a certain gradient is a leading gradient, by a method explained in the last paper. When the number of leading gradients found in this way is equal to the full number of linearly independent forms, as given by generating functions, then, as these are also the gradients giving irreductible covariants, the cornriant given by any other gradient is reductible. 31. It was pointed out in § 51 (Q.S.A., VII) that the covariant type given by a binary gradient 1' 1 A 0(1) A(!) A AJ131 • • • A A, 112

in seminormal units is the continued transvectant

607

490

A. YouNo

[Jan. 19,

and hence also the covariant of a single quintic given by the irreductible form is Hence we deduce : THEOREM

XX.

The continued transvectant

( ••• ((/, f )'>-• j)A• •• .f )'>-• has the leadiny gradient

A 0 AA, A,., .. . A,., according to a principal sequence, when and only u·hen it is in·eductible according to that sequence.

In the case of ternary forms the coefficients of the ground form require two suffixes for their expression. They are connected with the tableau by regarding the first suffix as giving the number of letters of the corresponding set which lie in the second row, while the second suffix gives the number in the third row. The addition of the last quantic in a concomitant type may be described as a two-indexed transvectant (in a manner already known) . And the result of this transvectant can be obtained by differential operators. In the same way for quaternary forms. a three-indexed transvectant is to be used. Using the word transvectant in this extended sense, we see that an irreductible concomitant given b~· a gradient is equal to the corresponding continued transvectant, no matterhow many variables there are. Further, Theorem XX is true for all cases when the multiple indexes of the transvectants and the corresponding multiple suffixes of the coefficients are inserted, and not merely for binary forms alone. IX. Binary forms . 32. For binary forms the product of two seminvariants is given by Theorem X. It is the case of the product of the form

It does not follow that all the terms in the product represent separate seminvariants; some of them will unite under the operation of fG . But when a leading gradient may be expressed as a product of two leading gradients the difference of the products will give us at once a reduction 608

1933.]

ON QUANTITATIVE SUBSTITUTIONAL ANALYSIS

491

for a later gradient. The case of the product of the form dealt with by the same theorem, gives us the transvectant (C .. , CtJ)•.

Incidentally, we learn at once from the result what is the leading gradient of this transvectant. 33. That exactly the same analysis may be applied to seminvariants as functions of differences of the roots has been already pointed out. (Q.S.A., VII, §53.) The results obtained here, so far as binary forms are concerned, may be interpre!.ed this way. A direct connection between the seminvariants expressed in terms of the coefficients and the seminvariants expressed in terms of the roots, which was suggested in the place quoted, will now be established. A leading gradient, which therefore represents an irreductible seminvariant, is considered, A 0 AA, ... A>-.; where, as always, ,\6 ;;?= ,\&.-- 1 ;;?: ,\6_ 2 .. .. . A tableau of 3-1 rows all beginning with the same column is written down, the first row being a 1 a 2 .. . aA,, the second a 1 a 2 ... a"•-•' and so on. Then the gradient may be looked upon as the symbolical product

The tableau as constructed yields a root product where each index is the length of the corresponding column. The two products be called reciprocal; one is always given by the rows of a tableau, the other by the columns of the same tableau. The symbolical products are arranged in a definite sequence, this determines a definite sequence for the reciprocal products, which will be called the reciprocal sequence.

,,.·ill

34. In a seminvariant expressed by substitutional forms, the letters of permutation are regarded as the symbolical letters and they appear in sets of n (the order of the ground form), each set representing one symbol; the equivalence of members of a set gives rise to the operator G, the equivalence of sets as a whole to the operator r. If the seminvariant is regarded as a function of the roots, the ground form is written as a product of n linear factors, each factor giving rise to one letter of permutation for each degree of the seminvariant. The letters again appear in sets of n 609

492

A. YorNo

[Jan. 19,

and the operator G is still required because for each coefficient ofthe ground form in the seminvariant the roots appear symmetrically. The seminvariant expressed in the roots is, in fact, of the same form as its expression in the coefficients, except that a new operator H must be introduced. This is the product of the symmetric groups (divided by the order o! in each case) of the letters which represent each individual root separately, there being one group for each letter. Thus the seminvariant

rGm,., 11 X becomes the function of the differences of the roots HrGm,., 11 X .

Conversely, we may start with a function of the differences of the roots r' G' m,., 11 X, and obtain the seminvariant in the coefficients H' r' G' m,., 11 X.

Here G' represents the fact that the letters really represent a function of degree o in the coefficients of each of the linear factors; r' represents the fact that the function is symmetrical in the different roots; and, finally , H' puts the letters in sets of n proper to the symbolical representation. The two forms of expression of a seminvariant are thus entirely complementary. To a leading gradient on one hand,

corresponds a first term on the other,

a,

where a 1 , a 2 , ... , are the roots of a binary o-ic. Anything which can be proved about the one can be interpreted in the other case. With this in view, the term leading gradient may be used with either meaning. The reciprocal of a leading gradient for any principal aequence ia a leading gradient in the reciprocal aequence.

35.

THEOREM

XXI.

Consider a leading gradient

the irreductible covariant ( •.. ( (f, f)A• f)As .. •f)A, 610

1933.]

ON QUANTITATIVE SUBSTITUTIONAL ANALYSIS

corresponds to it.

493

For n, the order of/, we have

The covariant exists for n = Aa, and hence in this case there is a function t/>>-. of the differences of the A8 roots, which is such that "2:.ef,,., is equal to the seminvariant in question. For every irreductible covariant of/ of this degree and weight there is such a function of the differences of the A.s roots. And the several functions I.ef,,., are all non-zero and linearly independent. ,vhen the order off is increased, the seminvaraints corresponding to the leading gradients g remain irreductible. The functions I.ef,1,.,, the same in form but with extension in summation, remain also non-zero and linearly independent. Hence a definite non-zero covariant may be obtained from g in the following way. The factors of/ are arranged in fixed sequence a 1• a2, ... an.· The product of the firstµ factors is written/,.. that of the next v-µ factors is written/•• _,,; then the sum for all permutations of the roots ~ A, when A,+ 1 is on a lower row than A,. And a{so Ag = A0 _, wlten Ai does 110/ lie 011 a lou•,ir mu, than A6 _., and A0 = Ag_ 1 + I when i\ 6 lies 011 a lower row. For a normal leading gradient X, it is obvious at once that P,NPX,, is not zero. THEOREM V-For atry leading gradient X, and a'!)' standard tableau F,, either

or or

(i) P,N,X

= 0,

(ii) X is a normal leading gradient corresponding to Fp,

where Y" is a normal leading gradient corresponding to the standard tableau F,, from which Y., is derived; s is some permutation and ~ is numerical. The sequence of quantics in X will be first supposed to be the fixed pre-arranged sequence, so that according to conditions (V) A,+ 1 > i\, always. Consider the weights in order commencing with A1 , if i\ 2 > A1 , A3 > A2 and so on, the conditions that X may be normal are being fulfilled. Let the sign of equality first appear in the case i\,+ 1 = A,, then unless A,+ 1 lies on a lower row than A,, X is still normal for F,. Let us then suppose that A,+ 1 lies in a lower row than A,, but not in the same column, otherwise P,N,X = 0. Then, since and

i\,+ 1

=

X = (A,A,+1) X,

l\,,

PPN,X

=

(A,A,+1) P,,N.,X,

where F,, is a standard tableau derived from F,, by the interchange of A, and A,+1• Now A,+ 1 lies in a higher row than A, in F,, and thus X is normal for F., up to this point. Proceeding thus step by step we find a standard tableau F,, which is such that

P,N,X

= sP.,N.,X,

623

SUBSTITUTIONAL ANALYSIS TO INVARIANTS

89

where s is some permutation, and X is a normal leading gradient for F .. at least so far as the letter A._, is concerned ; and, indeed, as far as the last letter is concerned, unless A. does not lie in a lower row than A0 _ 1 and A6 = A6 _, 1. In this case the last condition of the definition is not fulfilled. For this case it is necessary to consider the symbolical form Z from which the leading gradient was obtained. Since A6 A6 _ 1 is odd

+

+

where Z' is a form which has an increased value of AA + A0 _ 1 ; the theorem will be assumed true for such ti.irms when the total weight does not cxn·cci 1ha1 of X . Then P,,N.,X = - P N .. (A,._,A,.) X + :E ~sP,N,Y, 0

=-

(A,_,AA) P.,N.,X

+ ~ ~sP,N,Y.,

here F., is derived from F.. by the interchange of A 6 _ 1 and A,.. Thus, A 6 _, is raised in the tableau in changing from F., to F.,, and Xis normal for F.,. The theorem is thus true when the quantics in X are arranged according to the fixed sequence ; when this is not the case X=sY, where the quantics in Y are arranged in the fixed sequence. Then

PPNPX

=

P.NpsY

=

:E~s'P.. N..Y,

from the former case. Thus the theorem is always true. § 11. THEOREM VI-Every lrading gradient can be linearly expressed in terms of.forms obtained by permutation from PPN,XP ; where Xp is a normal leading gradient for the tableau FPfrom which P,NP is derived. Let X be a leading gradient ; then

X

=

:E TX

=

:E :E PNMX

= :E ~sP,NPX.,

by the last theorem. The number of different forms obtainable from P,NPXP by permutation, i.e., by left-hand multiplication with a permutation, is J.. .. •,.... •h, when the tableau Fp belongs to T ....,.... "A' Amongst the normal leading gradients corresponding to F there is one which is of special importance, the normal leading gradient of minimum weight. This is obtained by the rule that when A,ti is in a lower row of FP than A, 0,

A,+1

=

A,+

1,

and otherwise

A,+ 1 = A,,

For example, the standard tableaux of T 32 are

624

while

A1

= 0.

90

ALFRED YOUN(; ON THE APPLICATION OF

the corresponding weight tableaux for normal leading gradients of minimum weight are ( 0 0 I) (0 0 ( 0 I 2). ( 0 0 0) I 2 ' I I ' 1 2 1 1 '

1)

It will be seen at once that these correspond to the terms of the generating function It will be found that in every case the normal leading gradients of minimum weight have z"'J. .. •,.... •h (z) for generating function. § 12. Let 6 6 X = II A,. ,-,, Y = II A,. µ,.

z2J3 • 2 (z).

, -1

,-1

be both normal leading gradients corresponding to the standard tableau F, Y being the minimum leading gradient. Then ",+ 1 - ,-, > µ,+ 1 - µ,, and therefore A, +I - µ•+I > A, - µ, ; while Let ", - µ,

= v,.

Then Z

6

= ,-1 II A,.•,

is a normal leading gradient corresponding to the tableau for T 6 which has only one row. Moreover, T 6 Z6 is the general leading gradient for a single quantic; and these forms arc enumerated by the generating function [l] {[8] !}- 1 as described in §8. Thus the normal leading gradients for T ...... ...•h are enumerated by the generating function g [l] {[8] !}- 1 , where g is the generating function for the minimum normal leading gradient. But this we have found to be (§ 9)

z"'.f. ..... ....

h

hence

(z) [l] {[8] !}- 1,

= z"'J..,

a., . THEOREM VII-The generating fu11ction for the mtmmum normal leading gradients corresponding to T ...... ... •h is z"'J... ••. ..."h [z]. g

O.: , .. .

It is easy to sec that the lowest weight of the minimum normal leading gradients corresponding to T ... ... ... "h is given by the first tableau, that in which the first a. 1 letters lie in the first row, the next a. 2 in the second row and so on. In the corresponding weight tableau every element of the r' h row is r - 1, and the total weight is «2

+ 2«s + ... + (h -

1) ah=

i:r .

Thus the leading gradients are all identified in correspondence with the terms of the generating function. § 13. There is no difficulty in extending these results to irreducible perpetuants. GRACE* proved that all perpetuant types can be expressed in terms of those of the form

(VI) • 'Proc. London Math. Soc.,' vol. 35, p. 107 (1902) ; and GRACE and YouNo, "Algebra of Invariants," pp. 327-330 and Appendix IV (1903) .

625

SUBSTITUTIONAL ANALYSIS TO INVARIANTS where

"1 >

1,

"2 > 2, ...

"6-2

> 26- 3 ,

and the sequence of letters is fixed ; and of products of forms.

x-_ x) .,S-1

function for the forms (VI) is (1

91

The generating

1 6_ 1 •

Woon* demonstrated that this result is exact, or, in other words, that there can be no linear relation between the forms (VI) and products of forms. GRACEt proved further that for a single form where the letters are interchangeable the index conditions for the forms (VI) may be written

the generating function being (1 - x 2 ) (1 - x3) ... (1 - x6) ·

The argument used establishes at once that all perpetuant types may be expressed linearly in terms of products, and of forms (a,a2)•• (a2a3)'• ... (a6-~a6-1)'6- 2 (a6-1aa)'6- 1+•a,

where the quantics are arranged in fixed sequence, and and AA - A6_, = 0 or 1 ; and of forms obtained from these by permutation. leading gradient sequence C of this form is

The

The whole of the preceding argument may be applied to this result, the generating function for perpetuant types being multiplied in every case by x~6- 1 - 1 to give the corresponding generating function for irreducible perpetuant types. The normal leading gradient for an irreducible perpetuant corresponding to a given tableau is the same in form as that in the former case except that each weight suffix A, is increased by 2•- 1, except in the case of ).. 6 which is increased by 23 - 2 • III-APPLICATION TO BINARY FORMS OF FINITE ORDER

§ 14. The generating function for covariant types of substitutional form T ......... "h of the binary n-ic has been given in (I), § 1. This will be written

;:_"f"... •u ... "h

(;:_) ;

* • Prnc.

t

London l\fath. Soc.,' vol. I, p. 480 (1904). 'Proc. London Math. Soc.,' vol. 35, p. 319 (1903) and

626

GRACE

and YouNG, op. cit.

9'2

ALFRED YOUNG ON THE APPLICATION OF

it is an integral function of z, for it was obtained in the first place in the paper quoted in § 1 in the form

in which all the elements of the determinant are integral functions of z ; hence the function itself is an integral function of z. Comparison with J. .. .,...."h (z) gives the relation

[8] ! !"•..• ,....•h (z)

(VII)

=

h

[I] ./. ...,....•h (z) ,~ 1 [ex,+ n + 1 - r].,.

The highest index of z inf"....,. ... •h (z) must then be

t

t

1)

=

on using the notation and results of§ 6. Replace z by z- 1 in (VII) and multiply the result by is simply multiplied by ( - )H 1, hence

z(

1+

1t

+~(ex,

1) +~ex, (n + 1 - r) - ( 3

=

f ••11 a,. ... •1,. (z)

:E

1+

&+I)

~

+-'

an - 2CJ = O, the right-hand side

Br (z' - z9-•) •

From the generating function for covariants linear in the coefficients of each of 3 quantics of order n, we obtain-in the same way as for perpetuants-

[(It-~

]a= ~J. ..•..... •hz"'f"....,. .....,. (z) .

When T ...... ... •h, Tp., P, • ... Pi are conjugate, and their tableaux differ by interchange of rows and columns, the generating functions are not quite so simply related as in the case of perpetuants. There is a relation, as we proceed to show. As in § 6 we take h = 3, and use the value of (,, as found there. Then JI

Now

[r,, + n + 1 - r]p, = [n + 3]a [n + 2]a ... [n + 4 - "-a]a [n + 2 - exa] 2 [n + I - ex 3 ] 2 ••• [n + 3 - ex 2 ] 2 [n + 1 - ex 2 ] [n - ex 2 ] ••• [n + 2 - ex.] = [n + 3]., [n + 2]., [n + l] ... [n

+ r]., = Il [n + r -

where

+ s]

-- (-)•• z-(m+l-,) ..,-("':il) [ m

where Hence

«,

s=1

+ 1 + «, - 1·]. ,,

n= -m-2. IT [n

+ r]., = (-) z~ II [m + 1 + a, x = - (m + I) 3 + -c:s'. 3

r].,,

1;r -

627

SU.BSTITUTIONAL ANALYSIS TO INVARIANTS

93

The other part of J"... •,.... •h (z) does not contain n, and is the same as for the conjugate set as was seen in § 6 . Thus the transformation of n into - m - 2 which 6 II 1] 6 into (-)• ,-- (m+l)o ..,.--=-.,;..,-..,..._ [m clnnges [=--=-,-,-.-....,.:-

+

'

[IJ'- 1

,_

'

+ }]

[l]"-1

at the same time interchanges conjugate generating functions.

I V-TIIE

CHARACTERISTIC FUNCTION

§ 15. 111 the fourth paper on Quantitative Substitutional Analysis* the characteristic function 11, .. ,... ,.... .. ,, or the symmetric group corresponding to the representation T •..•,.... "h was introduced. The conception is due to ScttuR. The characteristic function is defined as the function II)

-

... . ,....•,, -

I;

Xe, • . ~,. ... #u

(s )~' (s2 )~' .. . (sn)~"' ...!

f31!f32! .. . r~,.! I

~

~

where x,,,. ,,,.... #,. is the characteristic of the permutation which has (3, cycles of r letters each, for the various values of r. It was shown in the paper referred to that when every cycle of degree r is replaced by the symbol s, then T ....,.... •,, becomes J.,...,.... "h 1ll ......,. ... ..,,, P,N,M, becomes 11> ..... , . ... "h and P,a,,N,lVf, becomes zero when r and s are different. THEOREM VIII-The characteristic Jimction 41 . .. . ,. ... "h is equal to I 3 of the symmetric group the symbol s, for a cycle o.f degree r is replaced by ( I - z•)- 1 , the replacement being made for all cycles, 8 ! a = (8) ! . This is verified without difficulty for 8 = I, 2, 3 ; it will be assumed true for all degrees less than 8. In the symmetric group we have first those terms in which the last letter b is not permuted, these form the symmetric group of degree 8 - I, then those terms in which b occurs in a cycle of two letters and so on. Let P, denote the sum of all terms in which b occurs in a cycle of r letters. Then the sum 3

of all terms is :E P,. ,=I

Consider P, when the r - I letters associated with b are fixed, we may have the remaining 8 - r letters permuted by any permutation of the symmetric group of degree 8 - r.

Also the r - I letters may be selected in (;

=D

may have (r - 1) ! different positions in the cycle relative to b. replacement of cycles is made _ (8 - 1) ! P, - [r] (8 - r] ! · And a = ~ (8 - I) ! _ _!J_ ,_, [rJ [8 - r) ! - (8) ! ' by the lemma, which proves the theorem. § 19. THEOREM XII8

I

.~

.,_

J. ( ) • z -

l: h i:

p

(,,)

Here ex, 13 are generic symbols, and 13 stands for 13 1, ~I

630

[S 1!

x~ [/3d (13:] ... [/3t]. ~~, .•• '3h

+ /32 + •.. + ~. = 0 j

where

ways and also

Then when the

ALFRED YOUNG ON THE APPLICATION OF

9()

and h/J is the number of members of the ~ conjugate class of the symmetric group which consists of operations having cycles of degrees ~1 , ~ 2 , ••• ~k ; x./•> is the characteristic of this class in the representation T •. By Theorem VIII the characteristic function for this representation is Cl

= •u a:,, ...

"'la

= I a,+s-, I•

Here the cycles are represented by variables s1 , s2 , .. . s0 and the characteristic function is at once given by this equation with the ordinary rules of algebra. If we write (1 - x')- 1 for s, we obtain .,+,-r

But by Theorem I, § 4,

z"' f. ..... ... "h (z)

=[a] ! j (ot,

=

[ot,

1

+s-

r] '.

+: _

r) ! j = [a] ! .= 8\ [a] ! ;

h/J 1 ·)•> s/J, s/J, ... s/Jk

_ 1 :E h [S] ! . /J 'f../J [~i] [~2] ... [~k] >

- al which was to be proved. THEOREM XIII-

To prove this we use the well-known results :E x./•>x./J "

.

= 0,

:E (x./•>) 2

= a !/h/J•

Multiply the identity of the last theorem by x./•> and then sum the result for all representations ot, we obtain the desired result at once. Equation (III), § 2, is a particular case of this theorem. § 20. The generating functions for binary forms of finite order may be treated in the same way. We begin with a lemma as before. LEMMA JI. cf, (n, S)

=.~ ~!]' [rn + r] [n + a 1

r] 6 _,

-

a [n

+ 8] = 0. 6

This identity is easy to prove directly for the values 1, 2, 3 of a, and all values of n ; it is also obviously true for n = 0, and all values of a. We proceed to establish a double induction. Now fa],= [a - I],+ z1-, [a - l],-1 [r] ; and [rn r] = [rn 2r] - z'.,.fi1',., + fi1,,,ti1'.,. + G>'.,.fi1',,,).

(ii)

1

na,p

(iii)

,,,).

(iv) When every transposition both of the a's and b's is changed in sign, the permutations of the compound group are all unchanged ; hence, if the suffix letters of the units be also taken to define the corresponding tableaux, and F,. be the tableau obtained by interchanging rows and columns in F,,

(v) When every transposition of the a's is changed in sign, but the transpositions of the b's are unchanged, we obtain np'a'

= I:,- (fi1,·,,·fi1 v,v + fi1 •.,.,fi1',., + fi1,,,fi1'•.••. + fi1v,ofi1,,,,.). 1

The last consideration enables us to write down without further enquiry the value I T &(ob) _- ~1-I ~ ti1,sti1 r',s' • 1

It is to be noticed that the ordersj,f' of conjugate representations arc the same.

635

SUBSTITUTIONAL ANALYSIS TO INVARIANTS

101

§ 24. Let r represent as before the operation of taking the sum of the S ! terms obtained by permuting the letters a in the operand. Then (XI)

= ..~v,,,T,T,',,,tl>''•.•

= ..~v, •. ,T.T/bl .

The equation (XI) will be written (XII)

§ 25. FROBENrns* discussed a closely allied problem, and his results come in very usefully here. Consider two linear substitution groups, which are representations of the same abstract group. Let (o:, f3 and

= ], 2, ... f)

(r, a= 1, 2, ... f')

be corresponding substitutions in the two groups. A third group is constructed by compounding the substitutions of the first two, in this the substitution corresponding to those written above is

This group is a representation of the same abstract group as a linear substitution group with ff' variables. In general, this third group is reducible ; let be its group determinant, then the reduction may be expressed by the equation

= n,. ,.,1••,.,

where the suffixes K, >-, µ define irreducible representations of the group, µ and µ' being conjugate imaginary representations (they are the same when the representation is real), and K, >- define the representations by the two groups from which we started. • 'S. B. berl. math. Ges.' (volumes are not numbered), p. 330 (1899).

636

102

ALFRED YOUNG ON THE APPLICATION OF

FROBENIUS proved that f.,,,_ is a positive integer or zero unchanged m value by any permutation of the suffixes. Its value is given by

hf.,,,.= l: x (R) x (R) x (R) ;

•

where his the order of the group, and x (R) is the characteristic of the group clement R in the representation K. Also J.J;. = l:f.,,,_f,,_, µ.

=

l:f.,,,_ 2

L h/hp, p

where hp is the number of operations in the pth conjugate class. § 26. The problem of FROBENIUS is not quite identical with that considered here, and so it is not permissible merely to quote his result without further enquiry. Let x (R) be the characteristic of the permutation R in the representation T. of the symmetric group of degree 8. When the permuted letters have to be expressed, we write R (•> or R (bl as the case may be, and for the compound group 0

Then from equation (XII)

= I: f.~1 x(S),

when R and S are not conjugate. Moreover, these equations are the necessary and sufficient conditions for the truth of (XII) . FROBENIUS found the equation

(XIII)

x· (R) x' (R)

=

~ µ.

f.,,,_, x("> (R'),

and from which he derived the value of J:,,,.. Multiply (XIII) by x. (S), and sum the equation for all values of 1., we have when S is not conjugate to R. and when S = R

o = .,,.l:

f.,,,. , x (S) x 6 (z) ;

for it is a function of degree 8 - 2 in x and iny and has a factor (1 - x) 6 - 2 , and also (l -y)'-2. Hence (1 - z) 2 xP._, (x, z) - (x - z) Pa (x, z)

Now put x

= z, 11>6

Thus

(x - z) P, (x, z)

=

(z)

= z (1

=

(I - x)'- 2

6

(z).

- z)-6+ 4 P,_1 (z, z}.

(I - z) 2 xPa-1 (x, z) - z (1 - x)a- 2 (I - z)-H 4 P6-1 (z, z),

gives a scale of relation to calculate P,.

641

SUBSTITUTIONAL ANALYSIS TO INVARIANTS

107

The following results are obtained : -

= 1, P4 (x, z) = 1 - xz, = 1 + z - 4xz + zx + z2x2.

p 3 (x, z) P, (x, z)

2

§ 32. The second term in.fs (x,y), due to X 5Y5 , is ( I - x) ( I - y) (x - y) x ( I - x5 ) ( I - _y') •

Then

{xy}' x (I - x) (I -y) (I - Z5 )

=

-y)

(xy}' (I

x LI - x

+ z (I

- x'1)

And

( 1 - x) ( 1 - y) (x - y) _ 1 - x x(t-xr•)(l-y'•) -

x3) + z2 x 2 ( 1 - x) (1-x5)(1-z6 )

x~. ,, Y~. ,, belongs to a

Herc we may use I - z•

In our case

11

=

= x" (l

1- y

+ z (I -

y3 )

+ z2y

l - y)

2 (

x ( l - y5) ( 1 - z5 )

•

very simple class, viz. : -

y

x -

X (

(1 - x)].

+ z( 1 -

- Y

The third term

+ z2x2

1 - x") ( 1 - y") .

- y")

+ y" (1

- x")

+ (1 -

x") (1 - y").

4 and the result is 1 - zx2 (I - x4) (1 - z4)

_

1 - .zy2

Y x (1 - y4) (I - z•) ·

The fourth and fifth terms may be taken together, for

( I - x) ( I -y) (x - y) (X 3 . ~ Y 3 . ~

+X

Y,.,,)/(x{[5] !}. {[5] !},) =2 (x - y) (1 + z) x (1 - x 2 ) (l - x'') (l - y 2 ) (1 - y3) ·

3 .,,

It is necessary to introduce the factors (1 - z3) (1 - z8 ) and the x, z function is 2 ((1 - x2) (1 - x3) (1 - Z3) (1 - ze)}- 1 X [1 - zx (l - x2)

The sixth term

x~..

I

Y~.. I is

+ z x (1

- x) - zs (1 - x2) - z8x3).

x-y

x (I - x 2 )2 (l -y2)2' and the corresponding x, z function is

I- z

642

z (l - zli

(1 _ x2) (1 _ z2)a ·

3

108

ALFRED YOUNG ON THE APPLICATION OF

Lastly, we have X 2 ,

13

Ya,

and

13,

x (1 -

with the x, z function

x-y x) 2 (1 - x2 ) (1 -y) 2 (1 - y 2 ) '

x2

(1 - x) 2 (1 - x2 ) (1 - z) (1 - z2 )

2x (1 - x) + z (1 - x 2) (I - x) 2 (1 - x 2 ) (l - z) (I - z2 ) 2

+

+ (1

1-z

- x2 ) (1 - z2)s ·

§33. Before giving the final result for the case 8 = 5. There is a general remark that can now be made clear about these functions. The form of result that is sought is as in § 'l:7 :( 1 - x) (1 -y) (x -y)fa (x,y) x {[8]!}. {[o] !},

=

(1 - x) (x, z)

=

x12z114> (x-•,

..t-1).

In consequence of this equation it is only necessary to give the terms up to Z8 the rest can be written down at once ; then 4> (x, z}

= l+z(xa+x'+2x6+xe-x8-xu-x1u+x12) +z 2 (x+2x 2 +x3+x') +z3 (-1 +2.,+2x 2+3x3+2x'-x8-x7 -xA+x 10 ) +z' (I +2x+3x 2 +2x3+x'-2x6-2x8-2x7 +x0 +x• 0+x 11 -x 12 )

+t6(-1 +2x+3x

2

+4x-'+4x4+x6-4x6 -5x7 -3x8-2x0 +x 10 +2x 11 +x' 2 )

+z8 (3+3x+3x 2 -2x'-5x5 -5x8 -3x7 -ti+x'+2x• 0+2x 11 -x 12 ) +z1 ( 1+ 2x+3x2 +~3x3-x'-3x 5 -6x8 - 7x1-5x8 -x9 +x• 0+2x 11 +3x 12 )

+z8 (4+2x+2x 2 -x3 -4x'-6x 6 -7:>.... -5x1-2ti+x•+x 10 +3x 11 +x12 )

+ ... +~•1x•2, 644

110

ALFRED YOUNG ON THE APPLICATION OF

VII-THE GENERATING FUNCTION FOR TERNARY PERPETUANTS OF A SINGLE FORM

§ 35. It has been proved* that if G, be a generating function for ternary forms of any particular type, and r 6 be the corresponding generating function for gradients (i.e., coefficient products), G6

=

(I -x) (I -_y) (l -_y/x) I\,

where the indices of x and y respectively refer as usual to the second and third weights of the form (or of the coefficients). We therefore proceed to consider the generating function for gradients of degree o of a single ternary form of infinite order. A gradient is a product

A,,,, ... A,,,,, each factor being a coefficient of the form, and the two suffices of the coefficient are its second and third weights. For gradient types we name the coefficients further, say, with a prefix, to define which of the o quantics has supplied the particular coefficients. For a single quantic, this fact is substitutionally expressed by the operator {1 A 2 A ... 6A}, the suffices not being here expressed as the prefixes alone are permuted. For binary forms, where there is only one suffix, the fact is equally well and more conveniently expressed by a permutation of the suffices. In ternary forms where there are two suffixes the permutation (,A,A) is expressed by the compound permutation (p,p,) (q, q,) of the suffixes. Thus in the ternary case we have to do with the compound symmetric group. We have seen§ 22 equation (X) that the compound symmetric group

T 6(pq) Let

-

-

"'1 ' ~ jti1,s ti1,,..

w,,P

be any gradient of the binary coefficients ,Apq, where by operation with fi1 1, we may obtain a gradient

fi1,,

is a unit of T., ., ...•,,, then

w,,P, and thus f distinct gradients of this type. The number of distinct gradients for this representation T., ., ... "h has been found to be given by the generating function

{[o] !}- 1 J., ...•hf.' ...•h (x) . x... Hence the generating function for ternary gradients of a single form of infinite order IS

I;

({[o] !}- 1)x ({[o] !}- 1)1 f., ... "h (x) .f., ... "h (y) X.,Y.,,

= {(8]

!}, -I {(8] !},-I}; (x,y).

• YouNG, 'Proc. l:.ond. Math. Soc.,' vol. 35, p. 431 (1933).

645

111

SUBSTITUTIONAL ANALYSIS TO INVARIANTS THEOREM

form is

XX-The generating function for ternary perpetuants of degree i> for a single

(I - x) (I -y) (x - y)fi (x,y) X

{[i>] !}, {[i>] !},-

§ 36. Let us now consider the results of the last section in respect to the generating function. The expansion is an infinite series in both x and y . The terms :,( y' where s > r do not really concern us, as the seminvariants, with which only we are concerned, never have the third weight greater than the second. In fact, the generating function proper is the part called above , 8 G _ (I - x) ,f, (x, z) _ 4 ([i>J!).tj,(z) -:E ,,,xz. Consider the expression of the perpetuant as a symbolical product, it essentially contains two kinds of factors (abu), (abc) ; then the index of x in the generating function is the number of factors (ahu), and the index of z is the number of factors (ahc) . When 8 = 2,

I

giving the obviously correct result. When i> = 3

Gi = - - . , I - x·

Here as in the general case the x factors in the denominator refer to (abu) terms which may be taken exactly as in the binary case, viz. :-

(a, a,u)'• (a 2 asuY• ... (aA-2aA _,u)'A- 2 (a,. 1a4u)~',- 1,

where

see §8. The denominator factor 1 - z2 here refers to a factor (a 1 a2 a 3 ) 2" numerator term zx3 corresponds to the form

and the

;

(XV)

there are thus two sets of perpetuants of degree 3, one (a1 a 2 a 3)2" (a, a 2 u)' (a 2 a 3 u) 2'• 2•;

and the other this set multiplied by the form (XV) given by zx3. When i> = 4 : G4

=

+ z (x" + x + xr, - x + z (x + x z (I - x + z (x x4 - x~ - xfi + x') - z~ (x\ + x6) + z~ (I -

{1

4

4

7)

2 -

X

646

2

2) -

3

+x

x2

-

x3

x2

-

x' - x4) -

6)

z7 x7}

{(1 - x 2 ) (I - x-1 ) (I - x4) (I - ,c: 2 ) (I - Z3 ) (I - z4)}· 1 •

112

ALFRED YOUNG ON THE APPLICATION OF

The terms independent of x are

I+ ;: ,

9

(l -z~) (1-z'} (I -z1 ) · It is easy to see that we may take the perpetuants corresponding to the denominator factors to be (a, a3 a 3 )~, (a 1 a 2 a 3 ) 2 (a, ai a,)~,

(a 1 a2 a 3 )3 (a, a3 a,) 8 • And that corresponding to the numerator term z9 to be (a 1 a2 a 3 ) 6 (a 1 a3 a,) 2 (a 1 a 3 a,).

The generating function assures us that all perpetuants of degree four, such that the second and third weights are equal, may be expressed as a sum of terms

(a 1 a2 a3 )'• (a 1a 2 a4 )'' (a 1 a3 a,)'•;

where ).. 1

=

2µ 1

+ 2µ + 3µ + 6~, 3

3

>. 2

= 2µ + 3µ + 2c, 2

3

>- 8

= t,

t

= 0 or 1.

It is useless to say anything about the other terms without a careful investigation of them, which has not yet been undertaken. VIII-GENERATING FUNCTIONS FOR TERNARY FORMS OF FINITE ORDER

§ 37. We concern ourselves with the generating function for gradients. THEOREM XXl-The generating function /•> for gradients of degree I> of the ternary n-ic is given by k o , /•> = l: hp n x~, ; fJ

,-1

where (3 defines the partition (3 1 , (3 2 , • • • r~k of I>, h~ is the number of members of the corresponding class of conjugate operations of the symmetric group of degree I>, and X~l•l When n

= 0,

1

x2~

f~

1

X'

y~

1

=

xl•H)~

yl"+2)~

x~

y~

I

1

X/ 0 > = I,

6(OJ

-

1,

and the theorem is a truism. We assume it true for all ternary quantics of order less than n. Consider a gradient

&

n A,,,,.,, the sum of the suffixes of each coefficient must

,-1 be equal to or less than n. Let each gradient be divided into two factors, let the first factor contain all the coefficients the sum of whose suffixes is n, and the second

647

113

SUBSTITUTIONAL ANALYSIS TO INVARIANTS

all the coefficients the sum of whose suffices in less than n. Let P, be the generating function for those gradients in which r of the coefficients have the sum of the coefficients equal to n. Then Pr -- Q , • "'.s-,(•-I) ' ,t,.

where Q, is obtained at once from the generating function for the gradients of degree r of the binary n-ic, i.e.,

Q,

=

(x"+I _ y•H) (x•+~

(x - y) (x 2

-y•+~) ,,. (x•+r _ y•+•) y 2 ) ••• (x' - y')

=

1!1:..

l:

r!

,

-

~ X(n+l)y, _ y, c..i

J==.l

-

X Y1) '

by Theorem XV. Here h/> is the number of members of the y conjugate set in the symmetric group of degree r. Thus we obtain 11> .S

=

L• P

r::::11.0

1

=

L• [

r=l>

h Vi JI1 (X l•> _ X l•-ll) ]

l;_Y_ y l'=I

r!

[

YJ

Y,

h C6-,> n, X , (n-1) ] L ..:..:L.__ y' (8-r)!J-I )'J

,

The generating function for binary forms can be treated in exactly the same way, and the same equation is obtained, where now

X. _ [(n #

+ I) ~]

-

by § 21 the result is

. '

[f~]

k

l: hp IT X,, . p

,-1

It is evident that there can be no linear relation with constant coefficients between k

the products IT Xr/•>; and hence the result is due to the values of the coefficients ,-1

h/>, h/-•>, and is quite independent of the form assigned to the functions X/•>; hence for ternary forms 4

=

s

l: hp

,~o

i

n ,._,

XP,

as was to be proved. § 38. It will be noticed that the generating function just obtained is the result of putting for every cycle of degree r in the symmetric group of degree 8 the function X, and then dividing by 8 !. The same methods give us the generating function for gradient types of degree 8, and of any particular substitutional form. The result is Tm:01u:M XX 11-T!tr. g1·11tmli11.i: .fimdi1111 1f>,. for gradient types, of de.i:ree 8 of ternary n-ics, of suhslil1ttim1al.fnrm T,, is

8!

648

_

=

l: h8 fJ

4

x/•l ,,.,,n

I

X 8,.

114

A. YOUNG ON SUBSTITUTIONAL ANALYSIS TO INVARIANTS

These results may be extended at once to quaternary and higher forms. quaternary forms we take

x~

1

x3#

y3~

z3-

1

x2,

y2-

z2~

x'

1

1

.I 1

-

z-

x1•+3l#

ynee (A) or ( B) int.his connection. The significance of these diagram,; as representations is the new element in the situation, and it is the Editor·s IJC'lieftha.t the ideas eontaincd in Young's later papt>rs Q.S.A. VII-IX when taken in conjunction with subsequent group t.heo1·etical developments will prove of importanee for the problem of Invariants.

652

A.

222

[June 14,

YOUNG

I . Oomplete set., ~f type8 wMrJ,, a,re linearly independent I. The expression (a~,a~• ... aM was introduced in Q.8.A. VII, p . 310, to denote the covariant obtained from the natmal substitutional unit which has ,\1 of the n 1 letters a 1 , ,\2 of the n 2 letters a 2 and so on, in the second row; and then this was modified to {a~1 .. .} by the relation

{a~1 ... aM = The relation (-)A•{a~a~• .. . aM

was then obta.ine.,' = ft• Then 1-1

K•'

= W-K2 =

"i:, {)'/ = (W-K2)-(W-K1-f2) = K6+(2-K2, ,-a

K1 1

= W-K2-"i:.f/ = K1+f>2-K2.

For here the positions of a 1 and (IV)

"i:,

a6

are exchanged.

Hence

(t>a+Pa) ({}•-1+P&-1) {aOa«saEa-Pa af•-•-P•-•a«,-J1+1p} {>3 •. • &,_1 1 2 8 '.. &-1 6

+ terms which belong to two of tho original sets and terms with a higher

index of a 1 •

e•-:;-

5. The following extension is fowid useful. the faotor

Consider equation (I) (§1);

1) on the right-hand side limits the value of s1 to two

ranges of values, the one from O to f 1, and the other equal to or greater than f 1+&1+1, i.e. equal to or greater than K1• At the moment all values of s, which are greater than Kt will be ignored. On the left-hand side of (I) r1 is limited to the range from O to {>1• We introduce on the left the term

r= 1

Kt

and the coefficient (

g,+::-rt) becomes (-

)''; on the right this will

bring a ooeffioient with the factor ( _ )«,H, (::)

= (_ )31+1 (;:) ,

the other factors being as before. Here st is of necessity equal to or greater than Kt and hence the only term considered now is s1 = K 1• Thus the term introduced on the left is equal to in value and opposite in sign to that already on the right; and on addition the appearance of the new K1 term on the left removes the old one on the right. 656

tt6

A.

[June 14,

YOUNG

Thus equation (Ill) becomes

+ (- )l~~ (fataua) ... (f'-f~~&--1) {a?-l,-1+:tcr a;,a:•-•• ... a~}

+... +(-)Et~ (f2+u2)

f2

••·•(ft-1 +0-1-1) (ft➔-1 +u1+1) ... (f•-1 +u•-1) f1-1 f1+1 fs-1

-[{a01at•2 ... al•-•a••} s-1 ,

al•-2-,,-2 a••-1 a••-,•- 1+1,-1} {ao1 at,-,. 2 ... 4-2 1-1 I

+

= terms which belong to two or more of the original sets terms which contain a letter o.f• for some r with Pr> KrIt will be noticed that equation (IV) leads to the particula.r case of (V) where t = I. Also that in the result the two sets of terms, i.e. the{} and f terms, are interchangeable as a whole as they should be. 6. It is useful to extend equation (I) somewhat. There may be other lettcn, present, with which the result is not primarily concerned, they will introduce other terms which arc sometimes needed. It is only necessary to introduce one such Jetter a, a.nu the results for any number are at once a.pparent. Then a1; in §1,

{aw-ira'• 1 2 . . . a'•-• ,-1 aoa"} 6

657

1951.]

ON QUANTITATIVE SUBSTITUTIONAL ANALYSIS

(IX)

227

and henco (VI)

['ii' t (f,+~,-r,)] {a~-:i:r a~, ... a~~l a~ a"} ,-2 r-o

(u!u,) ~ (-)"1(8 -::-l) {a~a;= ... n:~:a:-:i:•-•a"+•} 1) ]{a!a~2 ... a,!~: a:-:i:, a"}. = (- )10 [ :E ~ (- )"• (8 l>t

=

(-)'°:E

1

1- : ; -

+ (u+l) fl :E(ft+::-r') {af-:i:r-l a;• ... a~~f a~ a"H} +terms in

au+2,

etc.

II. First applications of the identt'.ty of §J. 7. Let us consider the covariant types of binary forms of order n, and of degree S. They may be written in the form

{ai• a:• ... a~}= 0, and it is supposed here that the indices are in descending order of magnitude. Then when K1+K2+ ... +K6 ~ 2w+c5-l we have c5 sets of forms defined by the presence of a;•, r = 1, ... , S, in tho defining gradient, such that no relation is possible between members of different sets. Let K~ ___., K2

then provided

= ... =

Ka

= n,

nc5-2w= R~8-l,

where R is the order of 0, 0 is irreductible sequence A provided {a:• ... a~} is an irrcductible type sequence A of degree S-1. This of course is already well known. When R < S-1, we may take K 1 = Kz = ... = Kn+z = n, and

KJi+a = K11 +4 = ... =Ka= n+l: then

:EK= 2w+S-2,

and equation 111 gives a relation between the first R+2 sets. Now this relation together with the reductions for types of lower degree gives all the relations fo1· these types as is shown by Theorems I and II. Let us consider the results in the simplest cases. 8. For invariants R = 0, and the result is (VI)

658

3 3 •-iao} form is irreductible sequence (A) then the g form is irreductible sequence (B) and vice versa, for any reduction of the{} form sequence (A) leads to a reduction of the g form sequence ( B) and the other wa,y round. Thus we know* that a.ll semi-invariant types irreductiblc sequence (B) can be expressed in the form

Hence when the right-hand side of (VI) is irreductible sequence (A), then or every difference tr+i ~ n-g,_ 1 • In fact in a.II cases the differences

-tr

g,_ 1 = n,

0, 2ga_ 1 -ts-2 -n, 3ts-1 -ta_3 -2n, 4g,_1 -t,_4·-3ri, ... , 3g,_1 -(8-l)n are in ascending order of 'magnitude. THEOREM III. The necessary and sufficient conditions that an i·nvariant type of degree 8 of binary forrns of order n rnay be irreductible are given by equation (VI) together with the reductions for covariants of degree 8-1.

It may happen that {}r

= g,+1-r• r = 2, 3, ... , 8-1. at,-1 a"} I t = {aol at• 2 •• • 4-1 6

Then the invariant

has pcrmutational properties defined by the substitutional form into which it may be put, i.e. "f->..,.PrNrl. The various terms are defined by the tableaux Fr, which give PrNr. When the letter a6 is dropped F 7 becomes P/, a tableau appropriate to the covariant given by the first 8-1 letters of J. In this case the letter a, must Le added to the tableau Jt\' in a position consistent with the fact that [1-(- )w (a1 a,)(a 2a,_1 ) • •• ] I= 0. The equatiuu Vl itself always introduces limitations on the introduction of a, into 11' / . Thus when -0·6 _ 1 is even 13 is unchanged by the operation (a,a6 _ 1 ), awl when &6 _ 1 is odd this operation changes the sign of J, and therefore abo of If. Other results involving a 6 _ 2 etc. arc obtained in the :mme way . 'l'hc:,;e remarks will Le illustrntc3 , ... which does not vanish is positive. Now f,_ 1 -0-2 = n-&6_ 1 -0-2 -1. Thus the condition series is n-8',_1 -t?2 - l , n-8'6_ 2 -t?3 , .. . , n-t? 2 -8',_ 1 and the differences cannot all be zero for if the first is zero the last is + I. The covariant Ct is subject to an equation which reduces

when the first of the differences 8-2 - [8 _ 1 , 8'3 - [6 _ 2 , is positive. And here the condition series is

THEOREM

order one

IV.

...

which does not vanish

Conditions for the reduction of the covariant type of {a.oI ats 2 ... at4-•a"'=Gt 6-1 6 1

are y-ivcn by t/u-. re,ltu:tions .for deyree o-1 torrtl11>r 11•itl,. the fact that Gt is red-uctible, when the first of lite dijfcrctu:1-.s

which does not mnish is positive. when the first of the differences

And [l +(-)t,-,(aH a6 )]Ce

1·8

reductible

wltich does not vanish is positive.

It should be remarked that for a covariant of odd order both n and S must be odd, and hence

660

A.

230

YOUNG

[June 14,

A reduction is obtained at once for (1 + (- )t•- 1 (a1 _ 1 a6)l 0 1 when 0 1 is reductible sequence (B) for then each term on the left when 0 3 is reduced will give by the use of (VII) only terms which arc earlier than 0 1. The multiplicity of terms on the right of (VII) prevents the similar conclusion being made for 0 3 ; in fact it is easy to give instances where it is plain that 0 3 is irreductible sequence (A), while the latest Ct on the right is reductible sequence (B). 10. Covariants of order two.

The natural course here is to take K 1 = Ka = K 6_ 1 = K 6 _ 2 = n in equation (III), every other K having the value n+ I. The terms of the set K 6_ 2 introduce the difficulty that the latest term is no longer

a" at,-1 al•-2} {ao1 ala 2 ... 6-2 6-1 I when n > [ 6 _ 1 > f6 _ 2 • Fortherearetermswith gradient factor a!~J-Paf•-s+P which may be later sequence (A). In this case we multiply the equation (III) by a numerical factor and sum as follows: (VIII)

=

(-)!!{ 11.u at• I

:: •••

at•-"a" a11-1-r:ia•1-J-!!+y2} 6-:J 6-2 6-1 6

where for There is a reduction here for that form or set of forms which has the second highest index with the lowest value. The possibilities are

n-y1-l, n-y2 -l, n-y3 -l, {}2 = n-,2•

661

1961.]

ON QUANTITATIVE SUBSTITUTIONAL ANALYSIS

231

(TX)

The latest term on the right, which is the term to ho re [ 2 + {._1 +I, and 2n ~ fa+ 2g,_1 + I. And also when n= f 2 +f,_ 1 +1 > t'a+f,_2 +1, and f 3 -2f2 =0 or I. III. A correction. 14. It is necessary to point out that Theorem XIX, Q.S.A. VIII, p. 489 is not true as it stands; it requires restatement. This ahm applies to Theorem XX. It is trne as was pointed out in §31, p. 48!), that the covariant type given by a. binary gradient Aio Af; ... Ai~ in scmi-norma.1 units is t,lie continued trausvectant

No proof was given iu the vlace there referred to. It foHows a.t once from the fact that in semi-normal units, a permutation of some of the first r letters, has no effect on the later letters; and in the result the position of the later letters is unaffected. Thus any property which can he expressed by permutational operations on the semi-normal form

AA 0 A1 • • • A,-1

664

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234

[June 14,

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ho1ds in the same way for the form 1 6 C6 = {All> A-,2 >... A-,-1 >A.,>}.

That is C, is a covariant of 0 8_ 1 and / 6 and hence must be a numerical mu1tip1e of (C8 _ 1 ,f6 )1'•. 15. With a given defining gradient G for types, the natura1 form {G}, the semi-normal form {G} and its equiva1cnt continued transvectant a11 differ from each other by forms given by an earlier defining gradient*. Hence the natural forms {G} in equations (III) or (V) may be replaced by the semi-normal forms or by the continued transvectants. Consider now the types of binary forms of a definite order n for which the defining gradient a~a~• ... a~•= Ab1> At~> ... Ai'} has at least one factor a:. It will be supposed that Aa ~ ,\6_ 1 ••• ~ ,\2 ~ 0. Then a modified continued transvectant will be introduced ; we may take this to be

V. The leading gradients sequence (A), and the defining gradients for irreductible types of degree 8 of forms of order n are the same. THEOREM

It is well known to be true when 8 ~ 4 . It will be assumed to be true for a11 degrees up to 8-1. Consider the covariant type

a>-•} = {G• 6-1 a>-•} C = {a01 a>-2 2 .. • a>-•-• 8-1 6 a ' As

~

.\,_1 •.•

~ A2 ~

0.

Then in p1ace of C we may consider the transvcctant (P6 _ 1 , at)>-,, where the leading gnuliN1I, of 1'6 _ 1 iH 0 6 _ 1 • When R the order of C iR grenter than o-2, a.nd .\6 > ,\ 6_1 t.l1c modified f,rn11svecf-,a11t

may be used, where 11 6 _ 1 is t,Jw covariant. of t.l1c form of ord ,\6 _ 2 the modified transvccta.nt

• Q.S.A. VIII, Theorem XVIII.

See also Q.S.A. VII, p. 328, §22.

665

1951.]

ON

QUANTITATIVE SUBSTITUTIONAL ANALYSIS

(IX)

235

may be used in the same way, provided R ~ S-2. Consider now the covariant types O .:,( R .:,( S-2 which have defining grndient,s in which the highest index is A,. They may be regarded as types of forms of order A,1 . Thus the presence of a higher index is impossible: such a gradient is zero, for a~•+i = 0. The leading gradient is alike in this and further its highest index is As, otherwise it would correspond t.o a covariant type of forms of or1ler A._ 1 , i.e. a covariant of the negative order R-8, which is impossil,le. The leading graditmt is then G3 _ 1 a~•, where G,_ 1 is a leading gradient of df'gree S-1. Now in equation (Ill) the defining gra(lients on the left a.re given by the sequence a 1 , a 2 , •• • , as on the right the first set is given by the sequence beginning a-s, a 3_ 1 , ••• , the SC'C(md hy the sequence as_ 1 , as, a,3 _ 2 , •.• and so on. In each case the leading gradi >-,., >-~. > "• > 0 and "• < 0. Note further that "• is summed from l to n to yield ., ; alao that. R = (2n+ l)a- 2n. (Ed.)

670

A.

240

[June 14,

YOUNG

For the cubic the order of (A~oA~,A~•) is

and hence ,\0

> .\3 ; since v = .\0 - . \3 , the inequality becomes

This iH alway:,; satisfied when n 1 > n 2 , ·i.e. when .\0 > ,\,+.\3 , for then -r = 1. Wlwu -n 1 = n:?, -i.e. wlwu .\0 = .\:?+.\:1> th~ form h(i written (a1a 2 )'° where a 1 , a 2 rcprc1;c11t the same (JUautic, this is zero when w is o .\2,

T

V

< .\2, T = 0 ;

V

~ .\2,

7 = 1.

'T

= 1;

For the quiutic the forms (af a~ a!f-r) arc considered: they a.re subject to the equation*

where

Y1 +r2+Ya = w-1 .

When y 2 = Ya and w is odd, there is a. relation

which giv(•:-; a reduction for (af-Y1aga;•) in terms of earlier forms provided y 2 > y 1 ; this relation only holds goo

,\1 ;

-r2 = I, v2 ~ ,\ 1 •

Other cases must be examined further.

VI. 'l'he binary catwnical forms, quantics of even order. 23. In the ordinary theory the canonical form of a binary form f of order 2n is f -- a 12n+ a 2n+ pall1• a 22• ••• a,.2• ., • • • + a,.2n+ 2z z r. and the condition that the form can be expressed as the sum of n 2n-th powers is that the catalecticant invariant Ao A1 Kn=

A1

Aa

An A,.+1

vanishes, so that p vanishes with K,.. Now it has been shown (Theorem VII, §10} that the covariants off may be expressed in terms of those which have a factor K,. and those for which the defining grndi ,\,,_ 1 +-\,t. 11 _

In the same way when the covariants can be expressed in a form with three letters a, as can always be done up to the sextic, the necessary and sufficient condition that the form is irreductible is that

where

'T

= 1:T,.,

and

w even

'T,-

=

w odd

'T,-

= 1,

1, when v,. > An-t;

when v,. ~ ,\n-l;

=0 -r,. = 0

'T,-

when v,. ~ ,\11_1 ; when v,. < ,\n---1 ;

where n is the order of the quantic considered. VII. l!'urther results from and extensions of Theorem VI I I. 25. When the sots K,. in Equation III, that is the sets whose defining gradient:,; lmvc one of the factors a;•·, r = 2, 3, ... , 8-1, represent earlier form:,; in the sequence choscu, than the sets K 1 , Ka this equation becomes

(X)

(- )'°(aiaa:• ... a!:-l a1) = (a~a(• ... af:.i'. aj•)+earlier terms.

Consider the case in which each letter represents a quantic which is Then let us define ,c,., r = 2, ... , n-1 as in §21 writing n+ 1 instead of 8 as the number of tmme power of the original quantic as in Section V.

673

J!J5J .)

ON

QUA:N'J'l'J'A'IIV J, SUBS'J]'J U'fJOJ\AL ANALYSIS

(IX)

243

letters iu wm, and keeping o1t:; the d('grec in t,he coefficients of the ol'iginal quantic. 'l'he form with Kn+i is taken to be O of §21; awl further the case n 1 = n,tt-i ii; to be c011sid ,\n-l · But when v, = ,\,,_i, and w is odd, the condition requires examination in general. It is well known that when R ~ S-1 the irreductibility of the form is unaltered by the introduction of the last letter. When ,\11 = 1 and ,\,._ 1 = 0 there can be constructed forms for which the condition R ~ S-1 is necessary as well as sufficient. For instance in the case of A~• A~• A!• A~, A,. where ,\0 > l, ,\,> 1, ,\.> 1, ,\1 > l, each Tis land v+T=S-1, and for odd weights and A4 = 0 = [ 2 and in place of it (A 08 A 36 A 56 ) should be included. The additional factors are obtained as the solutions of

The la.st equation is solved by A:i = :tft--l-4Y:i+:J:'h+ ::!ys+Yo,

t°a = !!2+Y1+t:i1:;+:J!fu+4Y1·

A~=.'11+.'I~·

t°1 =if:1+Y.1+!1,,+!1n-+·Y1·

'!'ht· factor:-; an,

The tirst live factors a.re leading gradients, and need not be discussed. The squares of the next t.lu·ading gradients, as also is the product of A 0 A 5 a.11

,\5 •

(/3 1 ) (A 0 A 2 A 5 ) when fa> 0, ,\4 = 0; and some ot.her cases. (a 2 )

(A 0 A 4 ) always when

f3 > 0

and ..\2 = 0 ; and some other cases.

(/32 ) (A 06 A 36 A 54 ) always when ..\2 +,\a >

,\5 +2.

(/33 ) (A 0 5 A 34 A 44 ) always when "6 > µ, 0 where µ, 0 iR the lowest index of A 0 for which (A~•Ai•A~•A!•) is a quartic covariant. Thus all gradients in which both A 2 and A 4 are present are excluded by (a 1 ). Then all gradients which have either A 2 or A 4 present are excluded unless f 3 = 0. The (fJ3 ) factor always ii; present when the indices are high enough unless "6 = µ, 0 and so this equality will first be assumed. With this condition the only forms to he considered for which ..\2+,\a = ,\5+ I are: where £ = 0 or I. The existence conditions are 3£-j-2 = 2µ, 3 +f3 +T2, ,\2-f-3e+ I= 2µ, 3 +f3 • When ..\2 > I them is u. factor (A 0 4 A 22 A 3 2 A 54 )=1 12 ; otherwise ,.\3 < 6 in either case. Thus in a.II cast•s there is a factor (fJ:!) or I 12 wht•n the indices a.re g1·eat enough. It remains to consider th ,\5 ancl hence ,\3 = 5, ,\5 = 4, .:\0 = .:\4 +6; and the only covariants are (A 06 Al A:;4 ), and the direct products of this mul (A 0 A 4 ) = i.

(b) .:\5 =3:

,\3 =2µ 3 +e,

.\o=,V+.:\2 +2µ 3 +2e,

A4 =0;

4(,\0 ' +.:\ 2 +µ 3 )+6e = e3 + 11. There iR a fact.or (A 0 A 2 A 5 ) unless ea= 0 and this is impossible. Otherwise =0, .:\0 =A'0 +.\ 4 +2µ 3 +e, 4(.\o'+µ 3 )+6e=e3 +r+9, l-'a~2. There is a factor (A 0 A~) unless !:i = 0 or ,\4 = 0. The conditions cannot be satisfied with ea= 0. Then the covariants to be recorded are only (Al A 34 Ai;3), (Aoe Aae As3). .:\ 2

(c) ,\5 = 2:

4(,\0 ' +.\2 +µ 3 )+6e = ea+s,

.:\ 2 +,\3 ~ -'s+ I = 3;

hence when ea= 0, ,\2 = 1, ,\3 = 2, and the covariant is (Al A 2 A 32 A 62 ). Otherwise 4(.\o' +µ:i)+ 6e = ea+-r+6 awl ,\:I~ :l; the only forms to he recorded a.re (A 04 A} A 52 ), (A 04 A 33 A 52 ).

e

There can be no solution 3 = 0 so ,\ 2 = 0 only need be considered. Then 4(,\0 ' +µ 3 )+6e = ea+-r2 +4, also ,\3 ~ 2. There is a solution ,\a= 2, f 1 = 0. -r2 = 0; it is (A 04 A 32 Ai A 5 ). Otherwise ,\~ = 0 and the solutions (A 0aA;i2A 5 ), (A 0 4 A 3'A 5 ) must Le recorded. The covariants which have one of the fl factors must now be considered. As a preliminary, we make a list of the covariants already obtained in (iv) attaching to each in a bracket the value of A 1 = 4(,\0 - , \4 - , \6 )+,\5 -2,\a; (A 06 A:15 A 5 4 ) (2), (A/A:i 4 A 5 3 ) (:l), (A 0 6 A}Ar,3) (:J),

(A 0 4 Aa4 Ar,2) (2), (A 0 4 A/Ar, 2 ) (4),

(A 0 3 A 2 A 3 2 A:;2) (2),

(A 0 4 A;i2A,.2A 6 ) (I),

(A 0 3 A 3 2 A 5 ) (5),

(Ao4 Aa" As) (5).

The factor (~ 1) doPs not, change the class under which the covariants at'