The Classical Groups: Their Invariants and Representations (PMS-1) 9781400883905

Table of contents :
Preface to the First Edition
Preface to the Second Edition
TABLE OF CONTENTS
Chapter I. INTRODUCTION
1. Fields, rings, ideals, polynomials
2. Vector space
3. Orthogonal transformations, Euclidean vectorgeometry
4. Groups, Klein’s Erlanger program. Quantities
5. Invariants and covariants
Chapter II.VECTOR INVARIANTS
1. Remembrance of thingspast
2. The main propositions of the theory of invariants
A. First Main Theorem
3. First example: the symmetric group
4. Capelli’s identity
5. Reduction of the first main problem by means of Capelli’s identities
6. Second example: the unimodular group SL(n)
7. Extension theorem. Third example: the group of step transformations
8. Ageneral method for including contra variant arguments
9. Fourth example: the orthogonal group
B. A Close-Up of the Orthogonal Group
10. Cayley’s rational parametrization of the orthogonal group
1L Formal orthogonal invariants
12. Arbitrary metric ground form
13. The infinitesimal standpoint
C. The Second Main Theorem
14. Statement of the proposition for the unimodular group
15. Capelli’s formal congruence
16. Proof of the second main theorem for the unimodular group
17. The second main theorem for the unimodular group
Chapter III. MATRIC ALGEBRAS AND GROUP RINGS
A. Theory of Fully Reducible Matric Algebras
1. Fundamental notions concerning matric algebras. The Schur lemma
2. Preliminaries
3. Representations of a simple algebra
4. Wedderburn’s theorem
5. The fully reducible matric algebra and its commutator algebra
B. The Ring ofa Finite Group and Its Commutator Algebra
6. Stating the problem
7. Full reducibility of the group ring
8. Formal lemmas
9. Reciprocity between group ring and commutator algebra
10. Ageneralization
Chapter IV. THE SYMMETRIC GROUP AND THE FULL LINEAR GROUP
1. Representation ofafinite group in an algebraically closed field
2. The Young symmetrizers. Acombinatorial lemma
3. The irreducible representations of the symmetric group
4. Decomposition of tensor space
5. Quantities. Expansion
Chapter V. THE ORTHOGONAL GROUP
A. The Enveloping Algebra and the Orthogonal Ideal
1. Vector invariants of the unimodular group again
2. The enveloping algebra of the orthogonal group
3. Giving the result its formal setting
4. The orthogonal prime ideal
5. An abstract algebra related to the orthogonal group
B . The Irreducible Representations
6. Decomposition by the trace operation
7. The irreducible representations of the full orthogonal group
C. The Proper Orthogonal Group
8. Clifford’s theorem
9. Representations of the proper orthogonal group
Chapter VI. THE SYMPLECTIC GROUP
1. Vector invariants of the symplectic group
2. Parametrization and unitary restriction
3. Embedding algebra and representations of the symplectic group
Chapter VII. CHARACTERS
1. Preliminaries about unitary transformations
2. Character for symmetrization or alternation alone
3. Averaging overagroup
4. The volume element of the unitary group
5. Computation of the characters
6. The characters of GL(n). Enumeration of covariants
7. Apurely algebraic approach
8. Characters of the symplectic group
9. Characters of the orthogonal group
10. Decomposition and X-multiplication
11. The Poincar Spolynomial
Chapter VIII. GENERAL THEORY OF INVARIANTS
A. Algebraic Part
1. Classic invariants and invariants of quantics. Gram’s theorem
2. The symbolic method
3. The binary quadratic
4. Irrational methods
6. Side remarks
6. Hilbert’s theorem on polynomial ideals
7. Proof of the first main theorem for GL(n)
8. The adjunction argument
B. Differential and Integral Methods
9. Group germ and Lie algebras
10. Differential equations for invariants. Absolute and relative invariants
11. The Unitarian trick
12. The connectivity of the classical groups
13. Spinors
14. Finite integrity basis for invariants of compact groups
15. The first main theorem for finite groups
16. Invariant differentials and Betti numbers of a compact Lie group
Chapter IX. MATRIC ALGEBRAS RESUMED
1. Automorphisms
2. Alemma on multiplication
3. Products of simple algebras
4. Adjunction
Chapter X. SUPPLEMENTS
A. Supplementto Chapter II, §§9-13,and Chapter VI, §1, Concerning Infinitesimal Vector Invariants
1. An identity for infinitesimal orthogonal invariants
2. First Main Theorem for the orthogonal group
3. The same for the symplectic group
B. Supplement to Chapter V, §3, and Chapter VI, §§2 and 3, Concerning the Symplectic and Orthogonal Ideals
4. Aproposition on full reduction
5. The symplectic ideal
6. The full and the proper orthogonal ideals
C. Supplement to Chapter VIII, §§7-8, Concerning
7. Amodified proof of the main theorem on invariants
D. Supplement to Chapter IX, §4, About Extension of the Ground Field
8. Effect of field extension onadivision algebra
Errata and Addenda
Bibliography
Supplementary Bibliography, Mainly for the Years 1940 -1945
Index

Citation preview

T H E C L A S S I C A L GROUPS

PRI NCETON L A N D M A R K S IN MATHEMATI CS AND P H Y S I C S N on-standard Analysis, by Abraham Robinson General Theory o f Relativity, by P. A.M. D irac A ngular M om entum in Q uantum M echanics, by A. R. Edmonds M athem atical Foundations o f Quantum M echanics, by John von Neumann Introduction to M athem atical Logic, by Alonzo Church Convex Analysis, by R. Tyrrell Rockafellar Riem annian Geometry, by Luther Pfahler Eisenhart The Classical Groups, by H erm ann Weyl Topology from the Differentiable Viewpoint, by John W. M ilnor Algebraic Theory o f Num bers, by H erm ann Weyl Continuous Geometry, by John von N eum ann Linear Program m ing and Extensions, by George B. D antzig O perator Techniques in Atom ic Spectroscopy, by Brian R. Judd

THE CLASSICAL GROUPS THEIR INVARIANTS AND REPRESENTATIONS

BY

HERMANN W E Y L

P R I N C E T O N U N I V E R S I T Y PRESS PRINCETON, NEW JERSEY

Copyright 1939,1946 by Princeton University Press Second edition, with supplement, published 1953 Copyright © renewed by Princeton University Press, 1966 Published by Princeton University Press, 41 William Street, Princeton, New Jersey 08540 In the United Kingdom: Princeton University Press, Chichester, West Sussex All Rights Reserved ISBN 0-691-07923-4 ISBN 0-691-05756-7 (pbk.) Princeton University Press books are printed on acid-free paper and meet the guidelines for permanence and durability of the Committee on Production Guidelines for Book Longevity of the Council on Library Resources Fifteenth printing, and first paperback printing, in the Princeton Landmarks in Mathematics and Physics series, 1997 Printed in the United States of America by Princeton Academic Press 3 5 7 9 10 8 6 4

In Memoriam

ISSAI SCHUR

PREFACE TO THE FIRST EDITION E ver since the year 1925, when I succeeded in determ ining the characters of the semi-simple continuous groups by a com bination of E. C a rta n ’s infini­ tesim al m ethods and I. Schur’s integral procedure, I have looked tow ard th e goal of deriving the decisive results for the m ost im p o rtan t of these groups by direct algebraic construction, in particular for the full group of all non-singu­ lar linear transform ations and for the orthogonal group. Owing m ainly to R. B rauer’s intervention and collaboration during the last few years, it now appears th a t I have in m y hands all th e tools necessary for this purpose. The task m ay be characterized precisely as follows: with respect to the assigned group of linear transform ations in the underlying vector space, to decompose th e space of tensors of given rank into its irreducible invariant subspaces. In other words, our concern is w ith the various kinds of “ q u a n titie s” obeying a linear transform ation law, which m ay be prepared under the reign of each group from the m aterial of tensors. Such is the problem which forms one of the m ainstays of this book, and in accordance w ith the algebraic approach its solution is sought for not only in the field of real num bers on which analysis and physics fight their battles, b u t in an arb itra ry field of characteristic zero. However, I have m ade no atte m p t to include fields of prim e characteristic. T he notion of an algebraic invariant of an ab stra ct group y cannot be form ulated until we have before us the concept of a representation of y by linear transform ations, or the equivalent concept of a “ q u an tity of type 81.” The problem of finding all representations or quantities of y m ust therefore logically precede th a t of finding all algebraic invariants of y. (For the notion of quantities and invariants of a more general character, and their close in ter­ dependence, the reader is referred to the restatem ent in C hapter I of K lein’s E rlanger program in slightly more ab stra ct term s.) M y second aim, then, is to give a m odern introduction to the theory of invariants. I t is high tim e for a rejuvenation of the classic invariant theory, which has fallen into an alm ost petrified state. M y vindication for having proceeded in a m uch more conservative m anner th an our young generation of algebraists would probably deem desirable, is the wish not to sacrifice the p ast; even so, I hope to have broken through to the m odern concepts resolutely enough. I do not pretend to have w ritten the book on m odern invariant theory: A system atic handbook would have to include m any things passed over in silence here. As one sees from th e above description, the subject of this book is ra th e r special. Im p o rtan t though the general concepts and propositions m ay be w ith which the m odern industrious passion for axiom atizing and generalizing has presented us, in algebra perhaps more th an anyw here else, nevertheless I am convinced th a t the special problem s in all their com plexity constitute the stock and core of m athem atics; and to m aster their difficulties requires on the whole the harder labor. The border line is of course vague and fluctuating. B u t quite intentionally scarcely more th an two pages are devoted to the general theory of group representations, while the application of this theory vii

PREFACE TO THE FIRST EDITION

viii

to th e particu lar groups th a t come under consideration occupies a t least fifty tim es as m uch space. T he general theories are shown here as springing forth from special problem s whose analysis leads to them w ith alm ost inevitable necessity as th e fitting tools for th eir solution; once developed, these theories spread th eir light over a wide region beyond their lim ited origin. In th is spirit we shall tre a t am ong others th e doctrine of associative algebras, which in th e last decade has risen to a ruling position in m athem atics. T he relations to other p arts of m athem atics are em phasized where occasion arises, and despite th e fundam entally algebraic character of th e book, neither th e infinitesim al nor th e topological m ethods have been om itted. M y experi­ ence has seemed to indicate th a t to m eet th e danger of a too thorough spe­ cialization and technicalization of m athem atical research is of p articu lar im portance in America. T he stringent precision attain ab le for m athem atical th o u g h t has led m any authors to a mode of w riting which m ust give th e reader an im pression of being shut up in a brightly illum inated cell where every d etail sticks out w ith th e sam e dazzling clarity, b u t w ithout relief. I prefer th e open landscape under a clear sky w ith its depth of perspective, where the w ealth of sharply defined nearby details gradually fades aw ay tow ards the horizon. In particular, the massif of topology lies for this book an d its readers a t th e horizon, and hence w hat p arts of it had to be tak en into the picture are given in broad outline only. An ad ap tatio n of sight different from th a t required in th e algebraic parts, and a sym pathetic willingness to cooperate, are here expected from th e reader. T he book is prim arily m eant for th e hum ble who w ant to learn as new th e things set fo rth therein, ra th e r th a n for th e proud and learned who are already fam iliar w ith th e subject and m erely look for quick and exact inform ation ab out this or th a t detail. I t is neither a m onograph nor an elem entary te x t­ book. The references to th e literatu re are handled accordingly. T he gods have im posed upon m y w riting th e yoke of a foreign tongue th a t was n o t sung a t m y cradle. “ W as dies heissen will, weiss jeder, D er im T raum pferdlos geritten,” I am tem p ted to say w ith G ottfried Keller. N obody is more aw are th a n myself of th e a tte n d a n t loss in vigor, ease and lucidity of expression. If a t least th e worst blunders have been avoided, this relative accom plishm ent is to be ascribed solely to the devoted collaboration of m y assistant, Dr. Alfred H. Clifford; and even m ore valuable for me th a n the linguistic, were his m athem atical criticisms. H er m a n n W eyl P r in c e t o n , N . J.,

September, 1938. N o t e . A reference to formula (7.6) [or to (3.7.6)] indicates the formula 6 in section 7 labeled as (7.6) in the same chapter [or in Chapter III respectively].

PREFACE TO THE SECOND EDITION T he photostatic process em ployed for the reprinting ruled out any appreci­ able changes which otherwise m ight have been desirable. B u t a new chapter containing Supplem ents, a list of E rra ta and Addenda, and a short Bibli­ ography for the years 1940-1945 have been added. Two of the supplem ents develop an altern ate and more d irect approach to some of the problem s in the theory of th e orthogonal and sym plectic groups dealt w ith in C hapters II, V and VI. Supplem ent C describes a particularly straightforw ard and powerful process for th e generation of invariants discovered by M. Schiffer, whereas supplem ent D applies th e “ m atrix m eth o d ” of C hapters I I I and IX to the splitting of a division algebra by extension of the ground field, w ithout the lim itation to norm al algebras and finite extensions. H er m an n W eyl P r in c e t o n , N . J .,

M arch, 1946.

TABLE OF CONTENTS PAGE

P reface P reface

to the to the

F irst E d it io n ................................................................................................................. vii S econd E d it io n .................................................................... ix C h a pter I IN T R O D U C T IO N PAGE

1. 2. 3. 4. 5.

F ield s, rings, ideals, p o ly n o m ia ls................................................................................................... V ector sp a c e ............................................................................................................................................... O rthogonal tran sform ation s, E u clid ean v ecto r g e o m e tr y ................................................... G roups, K le in ’s E rlan ger program . Q u a n titie s..................................................................... In varian ts and c o v a r ia n ts..................................................................................................................

1 6 11 13 23

C ha pter II V E C T O R IN V A R IA N T S 1. R em em brance of th in g s p a s t ............................................................................................................ 2. T h e m ain p rop osition s of th e th eory of in v a r ia n ts ............................................................... A. 3. 4. 5. 6. 7. 8. 9.

F irst M a in T heorem

F ir st exam ple: th e sym m etric g r o u p ............................................................................................. C a p elli’s i d e n t it y .................................................................................................................................... R ed u ction of th e first m ain problem b y m eans of C ap elli’s id e n t it ie s .......................... Second exam ple: th e unim odular group S L { n ) ........................................................................... E xten sion theorem . T hird exam ple: th e group of ste p tr a n sfo r m a tio n s................... A general m eth od for in clu d in g contra varian t ar g u m e n ts................................................. F ourth exam ple: th e orthogonal g r o u p ........................................................................................ B . A C lose -U p

of the

C.

36 39 42 45 47 49 52

O rthogonal G roup

10. C a y le y ’s ration al p aram etrization of th e orthogonal g r o u p ........................................... 1L Form al orthogonal in v a r ia n ts ........................................................................................................... 12. A rbitrary m etric ground fo r m .......................................................................................................... 13. T he in fin itesim al sta n d p o in t ............................................................................................................. 14. 15. 16. 17.

27 29

56 62 65 66

T he S econd M a in T heorem

S ta te m en t of th e prop osition for th e unim odular g r o u p .................................................... C a p elli’s form al c o n g r u e n c e............................................................................................................... P roof of th e secon d m ain theorem for th e unim odular g r o u p ....................................... T h e second m ain theorem for th e unim odular g r o u p ...........................................................

70 72 73 75

C h a pter III M A T R IC A L G E B R A S A N D G R O U P R IN G S A. 1. 2. 3. 4. 5.

T heo ry

of

F ully R ed u c ib l e M atric A lgebras

F u n d am en tal n o tio n s concerning m atric algebras. T h e Schur le m m a ....................... P r e lim in a r ie s............................................................................................................................................. R e p resen tation s of a sim ple a lg e b r a ............................................................................................. W edderburn’s th eorem .......................................................................................................................... T h e fu lly reducible m atric algebra and its com m u tator a lg e b r a ..................................... B.

T he R ing

of a

F in it e G roup

a nd

79 84 87 90 93

I ts C ommutator A l gebra

6. S ta tin g the p ro b lem ............................................................................................................................... 96 7. F ull r ed u c ib ility of th e group r in g ................................................................................................. 101 xi

TABLE OF CONTENTS

x ii

PAGE

8. Formal lem m as .................................................................................................................. 106 9. R eciprocity between group ring and comm utator algebra............................................. 107 10. A generalization............................................................................................................................. 112 C h a p t e r IV

T H E SY M M ET R IC GROUP A N D T H E FULL L IN E A R GROUP 1. 2. 3. 4. 5.

R epresentation of a finite group in an algebraically closed field ................................. The Young symmetrizers. A combinatorial lem m a ........................................................ The irreducible representations of the symmetric group................................................ D ecom position of tensor sp a c e ................................................................................................. Q uantities. E xpansion...............................................................................................................

115 119 124 127 131

C hapter V

TH E ORTHOGONAL GROUP A . T h e E n v e l o p in g A l g e b r a

1. 2. 3. 4. 5.

and t h e

O r thogonal I deal

Vector invariants of the unimodular group a g a in ............................................................ The enveloping algebra of the orthogonal grou p ............................................................... Giving the result its formal se ttin g ........................................................................................ The orthogonal prime id ea l........................................................................................................ An abstract algebra related to the orthogonal grou p ...................................................... B. T

he

137 140 143 144 147

I r r e d u c ib l e R e p r e s e n t a t io n s

6. Decom position by the trace op eration .................................................................................. 149 7. The irreducible representations of the full orthogonal group....................................... 153 C. T h e P r o p e r O r t h o g o n a l G r o u p 8. Clifford’s theorem .......................................................................................................................... 159 9. R epresentations of the proper orthogonal grou p ............................................................... 163 C h a p t e r VI

T H E SYM PLECTIC GROUP 1. Vector invariants of the sym plectic grou p ........................................................................... 165 2. Parametrization and unitary restrictio n ............................................................................... 169 3. Embedding algebra and representations of the sym plectic group............................... 173 C h a p t e r V II

CHARACTERS 1. 2. 3. 4. 5. 6. 7. 8. 9. 10. 11.

Preliminaries about unitary transform ations..................................................................... Character for sym m etrization or alternation alo n e......................................................... Averaging over a group............................................................................................................... The volume element of the unitary grou p ........................................................................... Computation of the characters................................................................................................ The characters of GL(n). Enumeration of cov a ria n ts.................................................. A purely algebraic approach..................................................................................................... Characters of the sym plectic group....................................................................................... Characters of the orthogonal grou p ....................................................................................... Decom position and X -m u ltip lication ..................................................................................... The PoincarS polynom ial............................................................................................................

176 181 185 194 198 201 208 216 222 229 232

TABLE OF CONTENTS

x iii

C h apter V III G E N E R A L T H E O R Y OF IN V A R IA N T S A. A lgebraic P art page

1. 2. 3. 4. 6. 6. 7. 8.

C lassic in varian ts and in varian ts of qu antics. G ram ’s th eo rem ....................................... T he sym b olic m e th o d .......................................................................................................................... T he binary q u a d r a tic ........................................................................................................................... Irrational m e th o d s................................................................................................................................ Side rem arks.......................................................................................................................................... H ilb e r t’s theorem on polynom ial id e a ls ...................................................................................... Proof of the first m ain theorem for G L (n ) ............................................................................... T he ad ju n ction a r g u m e n t..................................................................................................................

9. 10. 11. 12. 13. 14. 15. 16.

Group germ and Lie a lg eb ra s.......................................................................................................... D ifferential eq u ation s for invarian ts. A b solute and relative in v a r ia n ts................... T he Unitarian t r ic k ............................................................................................................................... T he c o n n e c tiv ity of the classical g r o u p s................................................................................... S p in o r s........................................................................................................................................................ F in ite in teg rity b asis for in varian ts of com pact g r o u p s.................................................... T he first m ain theorem for finite groups ................................................................................. Invariant differentials and B e tti num bers of acom pact Lie g r o u p ................................

B.

D iffe r e n t ia l

and

239 243 246 248 250 251 252 254

I ntegral M ethods 258 262 265 268 270 274 275 276

C hapter IX M A T R IC A L G E B R A S R E S U M E D 1. 2. 3. 4.

280 283 286 288

A u to m o rp h ism s....................................................................................................................................... A lem m a on m u ltip lic a tio n ............................................................................................................... P rod ucts of sim ple a lg e b r a s............................................................................................................. A d ju n c tio n ................................................................................................................................................ C hapter X SU PPL E M E N T S

A.

S

u p p l e m e n t to

C

h apter

II, §§9-13, a n d C h a p t e r VI, §1, V ector I nvariants

C

o n c e r n in g

I n f in it e s im

al

1. An id en tity for infinitesim al orthogonal in v a r ia n ts.............................................................. 291 2. F irst M ain T heorem for th e orthogonal g r o u p ....................................................................... 293 3. T he sam e for th e sym p lectic g r o u p .............................................................................................. 294 B.

S u pplem ent

to

C hapter V, §3, a nd C ha pter V I, §§2 and 3, C on c ern ing S ymplectic and O rthogonal I deals

the

4. A proposition on full red u ctio n ...................................................................................................... 295 5. T he sym p lectic id e a l...................................................*...................................................................... 296 6. T he full and th e proper orthogonal id e a ls ................................................................................ 299 C.

S u pplem ent

to

C h apter V III, §§7-8, C o n c e r n in g .

7. A m odified proof of th e m ain theorem on in v a r ia n ts........................................................ 300 D.

S u pplem en t

to

C h a pter IX , §4, About E x t e n sio n

of the

G round F ield

8. E ffect of field exten sion on a division alg eb ra ......................................................................... 303 E rrata and A d d e n d a ................................................................................................................................ B ib l io g r a p h y ................................................................................................................................................. S u pplem entary B iblio graphy , M ainly for the Y ears 1 9 4 0 -1 9 4 5 ................................. I n d e x ..................................................................................................................................................................

^07 ^08 314 ^17

CHAPTER I

INTRODUCTION 1. Fields, rings, ideals, polynomials Before we can s ta rt talking algebra we m ust fix the field k of num bers wherein we operate, k is the closed universe in which all our actions take place. I should advise the reader a t first to think of k as the continuum ofthe ordinary real or complex num bers. B ut generally speaking, k is any set of elements a, called numbers, closed w ith respect to the two binary operations: addition and multiplication. Addition and m ultiplication are supposed to be commutative and associative. M oreover, addition shall allow of a unique inversion (sub­ traction), i.e. there is a num ber o, called zero, such th a t a

o = a

for every a, and each a has a negative —a satisfying a + ( —a) = o. tion shall fulfill th e distributive law w ith respect to addition:

M ultiplica­

«(£ + y) = («£) + (« t), from which one readily deduces the universal equation a-o = o.

(1.1)

M ultiplication also is required to be invertible {division) w ith the one exception necessarily imposed by (1.1): there shall exist a unit e or 1 satisfying (1.2)

a -c = a

for all a, and every a except o shall have an inverse a ""1 or 1/ a such th a t a -a ” 1 = c. Were c = o, all num bers a would be = o according to (1.1) and (1.2); this degenerate case we once for all exclude by the axiom c ^ o. Any num ber a gives rise to its multiples a = la ,

a + a = 2a,

2a -f- a = 3a, • • • ;

here the integers 1, 2, 3, • • • are symbols of “ m ultipliers” ra th e r th an num bers in the reference field k. Two cases are possible: either all the m ultiples nc

(n * 1, 2, 3, • • •)

of the u n it c are o, or there is a least n for which m — o.In th e la tte r case th e integer n m ust be a prim e num ber p . Indeed for a composite num ber n = niWa (neither ni nor ni = 1) we should have n€ = ni €•712€ = o, 1

2

THE CLASSICAL GROUPS

and hence nic or n-it would equal o in contradiction to n being the least vanishing m ultiple of c. One distinguishes these two cases by ascribing the characteristic 0 or p to the field k . In a field of prime characteristic p the p-fcld of any num ber a vanishes: pa = p(ea) = (jpt)a = o. In a field of characteristic zero we can form the aliquot p art P = a /n of a with any integer n, i.e. a num ber p satisfying the equation nfi = a. Indeed this equation am ounts to ne-P = a ) and as the first factor n t is o the equation is solvable according to the axiom of divisibility. Hence our field k contains the subfield of the rational m ultiples of e: m c/n

(n a positive integer 1, 2, 3, • • • , m any integer 0, dbl, ± 2, • • •),

which is isomorphic to the field of ordinary rational num bers m /n and m ay be identified with it. To this m ost prim itive field of characteristic 0 we shall always refer as the ground field k, and our rem ark thus asserts the fact th a t any field k of characteristic 0 contains the ground field k. From now on we shall assume the reference field k to be of characteristic 0 w ithout m entioning this restriction again and again; wre shall not try to discuss any of our problems in a field of prim e characteristic. So even when we use the phrase “in an arbitrary field” or something sim ilar we m ean “ in an arbitrary field of characteristic 0 ” . If one omits the axiom requiring the existence of an inverse o f 1 one obtains the general notion of a ring rath er th an a field; only addition, subtraction and m ultiplication are possible in a ring. The classical example is the set of all integers. If a product ap of two elements of the ring never vanishes unless a t least one of the factors vanishes, th e ring is without null divisors. S tarting with a given ring R w ithout null divisors, we m ay form ally introduce fractions a /p as pairs of elements a, p i n R of which the second term P is ^ 0, and then define equality, addition, and m ultiplication in accordance with the rules which we all learned in school. T he fractions form a field, the quotient field of R; it contains R if we identify the fraction a/1 with a. W ith respect to a given ring R a set a of its elements is called an ideal if a d= P,

\a

lie in a for any a, p in a and any num ber X in R. The case where a consists of the one element 0 only is expressly excluded. The classical example is provided by the integral m ultiples of a given integer. T he ideals serve as modules for congruences: X ss y (mod a)

3

INTRODUCTION

m eans th a t the difference X — u of the two num bers X, m of R lies in a. A finite num ber of elements , • • * , a r in a constitute an (ideal) basis of a if every elem ent a in a is of the form Xi o!i -f- • • • -f- \fOtr

(X* m R).

a is then the ideal ( « ! , * • • , ar) with the basis ax , • • • , a r . In a field k there is only one ideal, the field itself. For if a is a num ber 5^ 0 in the given ideal a, th e latter will contain all num bers of the form Xa and hence every num ber 0 whatsoever: X = 0a~l. In the ring of ordinary integers every ideal is a principal ideal (a). a is a prime ideal if the congruence X/x 3= 0 (mod a) never holds unless one of the factors X, ju is = 0 (mod a). A formal expression f(x ) = 2 otiX* t-0 involving the “indeterm inate” (or variable) x, whose coefficients oti are num bers in a field fe, is called a (fc-) polynomial of x of formal degree n. If an 7* 0, n is its actual degree; 0 is the only polynomial not possessing an actual degree. Everybody knows how to add and m ultiply polynomials; they form a ring k[x] w ithout null-divisors. Indeed if a is of actual degree m, b of degree n: a = a mx m + • • - ,

b = 0nx n + • • •

(0Lm 9* 0, &n 7* 0),

then ab = a , | 8, i * +B + • • • is of degree m + n since amfin 0 . One sees th a t this proposition will still hold when the coefficients are taken from a ring w ithout null-divisors rath er th a n from a field k. This allows us to pass to polynomials of a new indeter­ m inate y w ith coefficients taken from k[x] or, w hat is th e same, to fc-polynomials of two indeterm inates x, y, and so on. The k-polynomials of several indeterminates x, y, • • • form a ring k[x, y, • • •] without null-divisors. In a given polynomial F(u, v, • • •) of certain indeterm inates u, vy • * • one m ay carry out the substitution « = /(* > y> •••),

» = g(?, y, • • ) , • • •

by means of certain polynomials / , g, • • • of other indeterm inates x } y, • • • ; th e result is a polynomial $(x, y } • • •) of x, y, • • • : y, ■■■), g(x, y, •■■), • • •) = $(x, y, • • •)• In particular one m ay substitute numbers a, 0, • • • for the “ argum ents” u ) v, in F ; th e resulting num ber F (a, 0, • • •) is called the value of F for the values a, 0, • • • of the arguments u, v, • • • .

4

THE CLASSICAL GROUPS

/Or) being a polynomial in a;, a is a zero or a root of / if f(a ) = 0 . A poly­ nomial of degree n has a t m ost n different zeros; this follows in the well-known way by proving th a t f(x ) contains the factors (x — ai)(x — a2) • • • if a x, ct%, are distinct zeros. Hence a polynom ial/(a;) ^ 0 does not vanish num erically for every value of x in fc, provided the reference field k is of characteristic 0, because such a field contains infinitely m any num bers. One can even find a rational value of x for which the value of / is 9* 0. Induction with respect to th e num ber of indeterm inates perm its generalization of this proposition to polynomials with any num ber of argum ents. If F(x, y, • • • ) ;

Ri(x, y, •••), •••

are a num ber of non-vanishing ^-polynomials then the product F R XR 2 • • • is also 0 ; and hence our statem ent can be sharpened to the following L e m m a (1.1.A). (Principle of the irrelevance of algebraic inequalities.) A k-polynomial F(x, y, • • •) vanishes identically i f it vanishes numerically fo r all sets of rational values x = a, y = ft • • • subject to a number of algebraic inequalities Ri(ot, ft • • • ) ? * 0,

R 2(a , ft • • •) 9* 0, • • • .

From the ring k[x, y 1 • • •] of ^-polynomials in x f y } • • • one can pass to the field k(x, y , • • •) of the rational functions of z, y, • • • in fe by forming the quotient field of k[x} y , • * •]• T he derivative f (x) of a polynomial f(x ) is introduced as the coefficient of t in th e expansion of f ( x -f 0 as a polynomial in t : (1.3)

f ( x + t) = f ( x ) + t-f'(x ) + . . .

The fam iliar formal properties of derivation are im m ediate consequences thereof. One m ight restate th e definition (1.3) as follows: there is a polynomial g(x, y) satisfying th e id en tity (1-4)

f( y ) - /Or) = (y - x)-g(x, y);

/'Or) is = g(x, x). While in Calculus the unique determ ination of g(x, x) is brought about b y requiring g(x, y) to be continuous even for y = x, Algebra attain s th e same b y requiring g to be a polynomial. The derivative of /Or) = ao -f- atiX -f- a2x -f~ * ** -h Gn%n is /'O r) = ai -f- 2a2x + • • • -h nanx n \ Hence the only polynomial f(x ) in a field of characteristic zero whose derivative /'O r) vanishes is the constant: f(x ) = a0 . F or a polynomial /(Or)) = f ( x i , • • • ? x n) of n variables one m ay form sim ilarly to (1.3): (1.5)

f{{x + t .y)) = f ( x x + tyL, • • •, x n + tyn) = /((x )) + L /i((x, y)) +

5

INTRODUCTION

T he coefficient/x((x, y)) of t in this expansion by t is called the polarized polynormal Dyxf of / ; it involves the new variables in a homogeneous linear fashion: I W - | £dX\ v » + • •• + jH rV ». dXn

(1.6)

Sometimes the new variables yi are designated by dx> and then the polarized form is called the total differential d f of /. The polar process has the formal properties of differentiation: D ( f + g) = D f + Dg, (1.7)

D (af)

=

a -D f (a a num ber),

D ifg )

= D f- g + f . D g .

T he degree of a monomial x 'l'x ? • • • In* of our n variables x \ , Xt , • • • , x„ is th e sum r = ri + r* + ••• + r n of the non-negative integral exponents n , • • • , r„ . E ach polynomial /((* )) is a linear com bination of m onom ials; if all these m onomials are of th e same degree r: (1.8)

/((* )) = 2

a ri...rnx T i l ■■■ x'n,

(r, + • • • + r„ = r)

the polynomial is called homogeneous or a form of degree r. In (1.8) th e sum extends over all sets of non-negative integral exponents rx , • • • , r n w ith the sum r. M ultiplication of all variables x * with a num erical factor X has the effect of changing (1.9)

/((* ))

into

Xr -/((x)).

A nother way of w riting such a form is this: n

(1.10)

/((* )) = 2 »-1

P (il,

■■■, *,)* * * « > ir)yilXi2 • • * Xir *

Hence the sym m etric m ultilinear form f ( x , y, • • • , z) corresponding to the given form /(w ) of degree r arises from / = f(u ) by complete polarization: DxuDyu * ’ * ^*v/(w) — 1 I ^ 1 0(il ) i%f ‘ * * f ir)x i \ y * * * %ir • This again shows its uniqueness. 2. Vector space The next fundam ental concept on which we m ust come to a common under­ standing right a t the beginning is th a t of vector space (in k). A vector space P is a k-linear set of elements, called vectors; i.e. a dom ain in which addition of vectors and m ultiplication of a vector by a num ber in k are the permissible operations, satisfying the well-known rules of vector geom etry.1 n vectors Ci, • • • , cn form a coordinate system or a basts if they are linearly independent, while enlargem ent of the sequence by any further vector would destroy this independence. Under these assum ptions every vector i is uniquely expressible in the form

(2 .1)

£

#1 Cl -f* • ** + %ntn

7

INTRODUCTION

where the num bers are the “components” of T he num ber n, which does not depend on the choice of the coordinate system, is called the dimensionality of the vector space P or the order of the linear set P. Transition to another coordinate system Ci, • • •, t n is effected by a non-singular linear transform ation A as described by the m atrix || a** || in the following m anner: £ = (2.2)

X\

+ • • • -j- x n cn — X\ t\ -f- • • • -j- x n tn \

e'i = 23

>

k

x * = 23 aikxk k

(i, k = 1, • • • , n).

A non-singular m atrix A — 11 a** 11 is one whose determ inant, det A or | A j , is different from 0 ; the inverse transform ation A ~ l sends the column of n num bers x' back into the column x. On w riting the com ponents in a column (m atrix of n rows and one column), (2.2) lends itself to the abbreviation x = A x'

(2.3)

in term s of m atrix calculus. There is another interpretation of this, or rath er of the modified equation x' = A x, to the effect th a t it describes a linear mapping of P upon itself in term s of a fixed coordinate system . In th a t case we need not suppose A to be non-singular. A m apping J —> j ' carrying each vector £ into a vector %' is linear if it sends £ + t) into

jf' + I)'

and

into

a |'

(a any num ber in fc). If such a correspondence changes the basic vector of our coordinate system into akitk >

Ci ^ k

it will carry | = ^ X iti

j ' = 23 Xit'i = Y ^ X i t i , i

intO

where (2.4)

x ’i = 23 aikXk

or

x' = Ax.

k

T he identical m apping %—» %is represented by the unit matrix E n = E = || M l W hen we express a given linear m apping $ —> (2.4), in another coordinate system in which the vector x has the components y given by (2 .5)

x = Uy,

U being the non-singular transform ation m atrix, the result will be y ’ = ( U~lA U )y ,

8

THE CLASSICAL GROUPS

as one readily derives from (2.5) combined with = Uy'

or

y ' = 1 F l x'.

Hence the m atrix A changes into U~~lA U which arises from A , as we shall say, by “ transform ation with U .” Therefore the characteristic polynomial | \ E - A | = \ n - h X * -1 + . •. ± bn of the indeterm inate X is independent of the coordinate system , in particular the trace b \ , trCA) =

»

and the determ inant bn = det A . A square m atrix A of n rows and columns is said to have degree n; the same term applies to any set 21 = {A } of n-rowed m atrices A . In the algebraic model of the n-dimensional vector space a vector simply m eans a sequence £ of n num bers: X = ( x i , • • • , x n). The num bers are the coordinates of %with respect to the “absolute coordinate system” : ei =

(1,

0,

•••

,0),

e* =* (0,

1,

•••

,0),

en =

0,

.. •

,1).

(0,

W hat our considerations have shown is the simple fact th a t every n-dimensional vector space in the general abstract axiom atic sense is isomorphic to this unique algebraic model. A linear form fix ) depending on an argum ent vector f m ay be defined w ithout reference to a coordinate system by the functional properties: f(X + f') = fix ) + /({')>

/(«*) = p". A group T is a set of correspondences containing the identity E f the inverse S ' 1 of any >S in T and the composite T S of any two correspondences S and T i n T. Considered as an abstract group y, our set T consists of elem ents s (of irrelevant nature) for which a composition st is defined satisfying the three rules: 1) the associative law (st)u = s(tu); 2) there is a unit element I such th a t I s = s I = s for all s; 3 ) every element s has an inverse s-1, ss~x = s~xs = I. W hen we tu rn to the abstract standpoint we shall always change the capitals like T, S, • • • into the corresponding lower case types y, s} • • • . T he given transform ation group T is a faithful realization of the abstract group scheme y. A realization of y is given if with every element s there is associated a one-to-one correspondence S: s S such th a t I -> E y

s~x -> ST1,

ts

TS;

it is faithful provided different elements s are associated with different S . Every group y in the abstract sense is capable of a faithful realization the point field of which is the group manifold y itself; this is accomplished by associating w ith the element a the “translation" ’ in y: (4.1)

(a): s' = as

(with the inversion s = a~V )

(regular realization). A realization by m eans of linear substitutions in an n-dimensional vector space is called a representation of degree n. This is not the place for repeating the string of elem entary definitions and propositions concerning groups which fill the first pages of every treatise on group theory.3 Following Klein's Erlanger program 4 (1872) we prefer to de­ scribe in general term s the significance of groups for the idea of relativity, in particular in geometry. Take Euclidean point space as an example. W ith respect to a Cartesian fram e of reference f each point p is represented by its coordinate x = ( x i , x2 , xz) f a column of three real numbers. (On purpose I deviate from th e common usage in calling the entire symbol (X i, x2 , x3) a coordinate, in the singular.) The coordinates are objectively individualized reproducible symbols, while the points are all alike. There is no distinguishing objective property by which one could tell ap a rt one point from all the others; fixation of a point is possible only by a dem onstrative act as indicated by term s like “ this,” “here.” All C artesian frames (of reference) are equally admissible; any objective geometric property possessed by one of them is shared by ail

INTRODUCTION

15

others. The coordinates x , x' of the same arbitrary point p in two such frames are linked by a transform ation S : (4.2)

x'i = a,• + X «■***

(*, k = 1 ,2 , 3)

k

where the non-singular A = ||

|| satisfies a relation

(4.3)

A* A = aE

with a num ber a (th at accounts for the arbitrariness of the yardstick). Each such transform ation (4.2) effects a transition from a given Cartesian fram e to another one. A t the same tim e S m ay be interpreted as the expression of a sim ilarity m apping p —> p f = p ' = x

or

x = x(p).

(The word “ field” is here used in the loose sense of a range of variability.) We suppose th a t the coordinatization sets up a one-to-one correspondence between the points p and the coordinates x. There is no objection to regarding as the fram e of reference this coordinatization itself. By m eans of an auto­ morphism a: p —> p' = S of the group y by means of one-to-one correspond­ ences x —> x' = S x w ithin th a t field. B. The “geometric” part (dealing with frames and quantities). (1) Any two frames f, f' determ ine a group element s, called the transition from f to f'. Vice versa, a group element s “ carries” a fram e f into a uniquely determ ined fram e f' = sf such th a t the transition (f —> f') = s. The transition f —> f is the u nit element I, the transition f' —> f the inverse element. If st t are the transitions f —» f', f' —> f" respectively, then the composite ts is the transition f -> f". (2) A qu an tity q of the type 21 is capable of different values. Relatively to an arbitrarily fixed fram e f each value of q determ ines a coordinate x such th a t q —> x is a one-to-one m apping of the possible values of q on the field of co­ ordinates. The coordinate x' corresponding to the same arbitrary value q in any other frame f' is linked to x by the transform ation x f = S x associated w ith th e transition (f —►f') = s by the given realization 21. F or a b etter understanding we m ay add the following remarks. The con­ nection between fram es and group elements as established by B (1) is very similar to th a t between points and vectors in affine geom etry.6 The last axiom under B (l) entails the associative law for composition of group elements. The epistemologist will stress the fact th a t the objects under A, the group elements and the coordinates, are objectively individualized and reproducible symbols, while any two fram es are, in Leibnitz’s words, “ indiscernable when each is considered by itself.” T hey are introduced in order to m ake possible the fixation of the values of all sorts of quantities in our geom etry by reproducible symbols. From a m athem atical standpoint one ought to observe th a t the axioms B (1) involve in no way more than the axioms defining a group, so th a t th e elements of every associative group m ay be considered as transitions

18

THE CLASSICAL GROUPS

between frames in an appropriate l'‘geom etry.'9 Indeed, if a group y is given, one m ay call each element s of y a t the same time a “fram e” and define the transition from fram e s to fram e t as the group element te-1. Then our axioms linking group elements to frames are obviously fulfilled provided the group m ultiplication is associative. In the same m anner, the m athem atician will not hesitate to identify the values of the q u an tity q with their respective coordinates x , and the requirem ent th a t only such relations m atter or have objective signifi­ cance as stay unaltered when x is replaced by *' = S x

(s -> $

in

21)

for every s will mean to him a mere convention by which he proclaims th a t he will study no other relations. All this sounds general and abstract enough. Nevertheless our form ulation B (2) is still too narrow for some im portant purposes since we have to consider the possibility th a t a single coordinate system will not be capable of covering the whole range of values of q. However, we are not going to dwell on such further generalizations; on the contrary, we shall from now on restrict ourselves to the particular case where the realization 21 is a representation and the co­ ordinate therefore any n-uple of num bers (x i, • • •, x n) in a given num ber field k (fc-vector). The word “ quantity" shall now be reserved for this case, and we once more repeat the definition under this lim itation :7 A quantity q of type 21 is characterized by a representation 21 of y i n k : s —> A(s) of a certain degree n. Each value of q relatively to a frame f determines a k-vector ( x i , • • • , x n) such that the “components'1 Xi of q transform under the transition s to another frame f' according to A(s). R epresentations of degree 1 are representations by numbers: s —> \ ( s ) ;

\(l) = 1,

\(st) = X(«)X(t).

T he particular representation of degree 1 for which X(s) = 1 identically in s: s —►1, is called the identical representation; a q u an tity having this type is called a scalar. We are now safely back in the w aters of pure m athem atics. The notions of inequivalence, reduction, and decomposition present themselves quite naturally in their application to a representation 21 of 7 or to a type 21 of quantities. They are of even more general significance inasm uch as any set of m atrices m ay replace the group 21. L et then 21 be a set of linear transform ations or m atrices A in an n-dimensional vector space P. If one changes the basis of th a t space by means of a non­ singular linear substitution U each A is changed into A ' = U~~lA U ; the A ' form the equivalent set 2 l'(^2 l). 21 is called reducible if P contains a linear subspace P' invariant under all the

INTRODUCTION

19

transform ations A of 21 th a t is neither the whole space P nor the zero space consisting of the vector 0 = (0, • • • , 0) only. W hen P is referred to a suitable coordinate system all m atrices A are then of the form (2.6). 21 decomposes if P breaks up into two non-vanishing subspaces Pi -f P2 invari­ an t under all A ’a of 21. In the coordinate system adapted to this decomposition Pi + P2 each m atrix A has the form (2.7), which fact shall be indicated by A — A\

A %.

These definitions apply in particular to a group 21: s i4(s) of m atrices A (s) homomorphic w ith the given group 7 . If U is a fixed non-singular m atrix the representation * -* A '(« ),

A '(s) = U~lA (s)U

is equivalent to the first one. We shall tre a t equivalent representations as one and the same, the difference lying only in the basis of the representation space in term s of which the linear operators are expressed in m atrix form. The trace x(s) of 4 (s ) is called the character of the representation. Equivalent representa­ tions have the same character. In case of reduction all A (5) have, relatively to the adapted coordinate system, the form (2.6). The p art Ai(s) defines a repre­ sentation 2Ii of 7 of degree m, and the p a rt ( x i , • * • , x m) of the q u an tity ( x i , • • • , x n) is itself a quantity, of type 2li ; we say th a t it is contained in the latter. If 21 is irreducible the q u an tity itself is called irreducible or primitive. In case of decomposition (2.7) our q u an tity consists of the juxtaposition of two independent p arts (X\ , • • • , Xni)t

(Xfn+1 , * • • , Xn)

whose components transform only among themselves. N othing prevents one from considering the electromagnetic four-potential together w ith the field strength as a single q u an tity of 10 com ponents; b u t of course this is a ra th e r artificial u nity and it is much more natural to decompose it into its two inde­ pendent p arts of 4 and 6 components respectively, th e potential and the field strength. I t obviously is of param ount im portance to know w hether a q u an tity breaks up into a num ber of independent prim itive partial quantities, i.e. w hether a given representation 21 m ay be split into irreducible constituents: is it true th a t a subspace Pi of P invariant under the operators A (s) of 2t possesses a comple­ m entary invariant subspace P2 such th a t the whole representation space P breaks up into the two linearly independent parts Pi + P2—and is this true for any representation 21 of the given group 7 ? T he answer is affirmative in the m ost im portant cases, in particular, as we shall see later, for all finite groups. On our way we encountered the following process of addition by which two representations 21: s —» A(s)

and

21': s —> A '(s)

20

THE CLASSICAL GROUPS

of the same group and of degrees m and n respectively give rise to the representa­ tion 21 + 21' of degree m + n: s —>A(s) 4- A'(s). T he qu an tity (a?i, • • • , x m) of type 21 is combined with the quantity (yx, • • • , y n) of type 21' to form the qu an tity 0^ 1

j

’ * *

y Xm y Vl

f

’

j 2/ n ) *

The character of 21 + 21' is the sum of the characters of 21 and 21'. Another im portant procedure is th a t of multiplication 21 X 21'. If the vectors x = (xi f •• - ,Xm)y

y = (t/i, • • • , y»)

undergo the linear transform ations 4 = || dik ||

,

A ' = || a

( h fc = 1, **‘ \p , q = 1, • •. , n /

respectively, then the mn products (4.4)

Xiyp

(i = 1, •

p = 1, . • • , n)

, m;

undergo a corresponding linear transform ation A X 4 ', called the Kronecker product of A and A '. In explicit form C = A X A ' = || ci;Pl*, || is obviously given by / Cip ,k q

Q'ikQ'pq y

and our definition a t once yields the law of composition (A X A ')(B X B f) = (AB X A fB f). 21 X 21' is the representation s - ^ ( A ( s ) X A' W) of degree mn. The same sign X will be applied to the corresponding quantities. The character of 21 X 21' is the product of the character of 21 with th a t of 21'. The problem arises quite naturally to decompose the product of two prim itive quantities into its prim itive constituents, partial cases of which will be discussed later (C hapters IV and V II). T he num bers (4.4) m ay be considered as the components zip of a vector z = x X y in an mn-dimensional vector space PP'. W hen considering linear forms in th a t space it will often be convenient to replace the m ost general vector z with mn independent com ponents by the vector x X y where Xi and yP are independent variables; this procedure is called the symbolic method in invari­ a n t theory. Let us call attention right here to some im portant representations of the full

21

INTRODUCTION

linear group GL(n), by stressing the representation aspect of forms of n variables and their transform ation under the influence of linear substitutions on the variables. Take any non-singular linear transform ation A : = 52 aikx k . k

(4.5)

Under its influence all monomials of given degree r, x ? x r22 • • • x rnn

(n + r2 + • • • + r n = r),

undergo a linear transform ation (A)r , and this correspondence A —» (A ) r is a representation of degree n(n + 1) • • • (n + r — 1)/1 2 • • • r. On the other hand we consider an arb itrary form / of degree r depending on a contravariant argum ent vector ( f t , • • • , ft) and write it as (4-6)

/ = £ ri! r - r•^• •Vrn! ,

r j? •••

While ft- are transform ed according to (4.7)

ft- = 52 k

•

f changes into a form of the new variables ft- whose coefficients u Tl...rn proceed from uTl...rn by the same substitution (A )r as encountered above. Indeed, the rth power of the invariant product (ft;)

=

ftZ l +

• • * +

ftZ n

equals the special form (4.6) with Wri •••»•» — Xi • • • Xn . In the symbolic m ethod one replaces the arb itrary form / by the specialized (ftOr. The products of the components of r vectors x, y y • • • , z : Xiiyi2 * * * zir which are cogrediently transform ed into vectors x', y • • • , zf by the same A , (4.5), undergo the transform ation A X A X • • • X A . A q u an tity F of this type with the n com ponents F ( i i , i2 , • • • , i r) is called tensor of rank r. Ac­ cording to (1.10) we write our form (4.6) as X) v(ii i f - i r)$n

• • • £ viz. not to destroy congruence, am ounts to the require­ m ent th a t S commutes w ith the whole group 0 + of congruences: S~lA S « A ' m ust be a proper orthogonal m atrix whenever A is such. All linear trans­ form ations S satisfying this condition form a group, the so-called normalizer of 0 +. The normalizer of a group comprises th a t group; in our case it is actually larger because it contains the “ reflections” (improper orthogonal transform a­ tions) and dilatations, besides the “ rotations” . The group 0 + plays its intrinsic p a rt in Euclidean geometry long before the “ extrinsic” question of all auto­ morphisms arises; its normalizer rath er th an 0 + itself is the group of auto­ morphisms. W hat shall we say then to K a n t’s discussion of the distinction of “ left” and “ rig h t” in §13 of the Prolegomena8 where he claims th a t “ by no single concept, b u t only by pointing to our left and right hand, and thus de­ pending directly on intuition (Anschauung), can we make comprehensible the difference between similar yet incongruent objects (such as oppositely winding snails).” No doubt the m eaning of congruence in space is based on intuition, b u t so is similitude. K an t seems to aim a t some subtler point; b u t ju st this

INTRODUCTION

23

point is one which can be subsum ed under the general “ concept” of a group and its normalizer. While a group is, generally speaking, not derivable from its normalizer, there is nothing m ysterious in the possibility th a t the norm alizer m ay be actually larger th a n the group itself.

5. Invariants and covariants L et there be given a group T of linear transform ations A in an n-dimensional vector space P. A function /(x , y , • • •) depending on a num ber of argum ent vectors (5.1)

x = (x x, • •. , xn),

y = ( y i , • • • , y n), • • •

in P will change into a transform / ' = A f if x, y, • • • are sent by the linear transform ation A into x' = Ax, y f = A y, . . . : /'( * ', y f, • • •) « /(« , V, • • •)• We then have B (A /) = (BA)/, as it should be. If A f = / for all substitutions A of our T, the function / is called an invariant of T. In this sense the scalar products (xx), (xy ), • • • are orthogonal vector invariants. We shall be con­ cerned with the algebraic case exclusively where / is a polynomial, homogeneous with respect to the components of each argum ent vector, and therefore is called an invariant form. The degrees n, v, • • • of / in x, y, • • • m ay coincide or not. W ith this elem entary notion we contrast the general notion of invariant. While the former is concerned with a given group T of linear transformations A, the latter is relative to a given abstract group y = {s} and a num ber of repre­ sentations of 7 of degrees m ,n , • •. respectively: « : s -> A(«),

(5.2)

33: 8 -+ B (s), . . . .

A function U(s) of degree N . The invariant forms /(# , y % • • •) are thus turned into linear invariants of a single quantity z of type U. However, one should emphasize the fact th a t this line­ arization of the problem of invariants is possible only if we study invariant forms of pre-assigned degrees y, v, • - • . T he linear invariants L{x) of a quan tity x = f a , • • • , z n) of given type 21: s —> A(s) form a subspace of the n-dimensional space of all linear forms of x. I being its dimensionality, we have exactly I linearly independent linear invari­ ants. On taking them as the first I coordinates of an arb itrary £ in a new co­ ordinate system, we obtain a reduction of 21 to the form A (s) =

Et '*

° ||. *

I is the maxim um degree with which the unit representation s —>Ei is contained in 21. W hen in particular the theorem of full reduciblity holds, we m ay describe I as th e m axim um num ber of times the identical representation s —> 1 is con­ tained in 21. An invariant m ay be described as a scalar depending on a number of arbitrary quantities x, y, ••• of prescribed types. If the transform sf differs from / by a constant factor X(s), (5.5)

s /= X W - /,

/ is called a relative invariant with the multiplier \{s). / = 0 is still an invariant relation between the variable quantities x ) y, • • • . The invariants in the original sense are then to be distinguished as absolute invariants. The m ultiplier is a representation s X(s) of degree 1. Still more generally, a covariant of type s —>H (s) is a q u an tity / of th a t type depending on argum ents x, y, • • • which are quantities of given types 21, ©, • • • respectively. W ith regard to a given fram e of referen ce,/ will have h com ponents f i = fi(x , V, • • • ) , • • • ,/a = fhix, y, • ■■), ju st as x 1 y } • • • have the com ponents (5.3). After transition 5 to another frame, the new com ponents which arise from the old ones by the linear sub­ stitution H(s) shall be the tran sfo rm s/! = s f i , • • • , / I = sfh ; hence on putting y', • • •) = fi(x, y, • • •)

(i

== 1, • • • , ’ft)

with x' = A(s)z,

y r = B (s)y, • • • ,

the equation / ' = sf = H (s)f is to hold.

The system of sim ultaneous equations fi

= 0, . . .

J h

= 0

then has an invariant significance, independent of the fram e of reference.

26

THE CLASSICAL GROUPS

As an illustration of relative invariants let us consider the classical case where y is the full linear group C?L(n), consisting of the linear transform ations A r x% =

k

ausXk ,

and where the quantities which appear as argum ents in / are arbitrary forms of given degrees in th e n variables . Under these circumstances X(A) will be a homogeneous polynomial of the n 2 variables ; hence for the special transform ation (5.6)

x'i* axi

the m ultiplier X(A) will equal a0 with a non-negative integral exponent G. On applying the relation \ ( A ) \( B ) = \{ A B ) to the transform ation A and th a t one B = || Aki || whose elements consist of the minors Ak% of A : A B = BA = A E where A = | A | is the determ inant of A , one finds (5.7)

H A ) \( B )

= A°

B u t since A is an irreducible polynomial of the n variables aik and X(B) is a polynomial no less th an X(A), (5.7) forces X(A) to be a power of A: (5.8)

\( A )

= A°

The integral exponent g is called the weight of the relative invariant. On account of formula (5.8) relative invariants of the full linear group GL(n) are absolute invariants of the unim odular group SL (n). On the background of these general notions concerning fields, vectors, groups, representations we now set out to study the algebraic vector invariants of the m ost im portant groups, in particular of the full and of the unim odular linear group, GL(n) and SL (n), and of the orthogonal group, 0(n ) or 0 +(n), in n dimensions.

C H A P T E R II

VECTOR INVARIANTS 1. Remembrance of things past The theory of invariants originated in England about the middle of the nineteenth century as the genuine analytic instrum ent for describing configura­ tions and their.inner geometric relations in projective geometry. The functions and algebraic relations expressing them in term s of projective coordinates are to be invariant under all homogeneous linear transform ations. Cayley first passed from the consideration of determ inants to more general invariants. This procedure accounts for the title of his paper, M^moire sur les H yperd6term inantsx (1846), which one m ay look upon as the birth certificate of invariant theory. In his later nine fam ous M emoirs on Q uantics2 (1854-1859) he succeeds, am ong other things, in obtaining a complete set of invariants for cubic and biquadratic forms. His work was taken up in E ngland by Sylvester and Salmon. Sylvester tau g h t a t Johns H opkins U niversity for some years, and there founded the first m athem atical journal on this continent: The American Journal of M athe­ matics. The pages of its first volumes are filled with papers on invariant theory from Sylvester's prolific pen. In Germany, Aronhold, Clebsch and Gordan became adherents and prom oters of the new discipline. In Italy, Brioschi, Cremona, Beltram i, and Capelli were attracted to the subject. This early period has a formal character throughout: the development of formal processes and th e actual com putation of invariants stand to the fore. Almost all papers refer to one group, the continuous group of all homogeneous linear trans­ formations. Another impulse, in a somewhat different direction, came from num ber theory, more particularly from the arithm etic theory of binary quadratic forms. H ere one had been led to consider not a continuous b u t a discrete group, the group of unim odular linear substitutions with integral coefficients. Gauss, in his Disquisitiones arithm eticae, studied equivalence of quadratic forms with respect to this group. Besides and after Gauss, we have Jacobi in Germ any and H erm ite in France as outstanding m en in this line of investigation. The formal period of classic invariant theory is followed by a more critical and conceptual one which solves the general problems of finiteness less by explicit com putations th an by developing suitable general notions and their general properties along such abstract lines as have lately come into fashion all over the whole field of algebra. Here there is only one m an to m ention—H ilbert. His papers (1890/92) m ark a turning point in the history of invariant th eo ry .3 He solves the m ain problems and thus almost kills the whole subject. B ut its 27

28

THE CLASSICAL GROUPS

life lingers on, however flickering, during the next decades. A. H urw itz m akes a new and im portant contribution by introducing integral processes extending over the group manifold (1897); in England A. Young, working more or less alone in this field, obtains far-reaching results on the representations of the sym m etric group and uses them for invariant-theoretic purposes (1900 and later). In recent times the tree of invariant theory has shown new life, and has begun to blossom again, chiefly as a consequence of the interest in invarianttheoretic questions awakened by the revolutionary developments in m athem ati­ cal physics (relativity theory and quantum mechanics), bu t also due to the connection of invariant theory with the extension of the theory of representa­ tions to continuous groups and algebras. T he rise of projective geom etry m ade such an overwhelming impression on the geometers of the first half of the nineteenth century th a t they tried to fit all geometric considerations into the projective scheme. The narrowing down of the projective group to the affine group or to the group of Euclidean motions of m etric geometry was accordingly effected by adjoining some so-called “ absolute” entities: the plane at infinity, the absolute involution. T he same attitu d e is expressed when one treats m etric geom etry in vector space by allowing arb itrary affine coordinate systems and arbitrary linear transform ations, while adding the fundam ental m etric form x\ + x\ + • • • + x \ as som ething absolute, instead of sticking to th e m etrically equivalent C artesian coordinate systems only and the corresponding group of orthogonal transform ations. As this procedure easily adm its extension into infinitesimal geometry, it has remained in use w ith great success, particularly for the purpose of general relativity theory. In group theory it am ounts to considering each group of linear transform ations as a subgroup of and in relation to the to tal linear group. T he dictatorial regime of the projective idea in geometry was first successfully broken by the Germ an astronom er and geometer Mobius, b u t the classical docum ent of the democratic platform in geometry establishing the group of transform ations as the ruling principle in any kind of geom etry and yielding equal rights of independent consideration to each and any such group, is F. K lein's Erlangen program. The adjustm ent of invariant theory to this standpoint has been slow; it could not be made w ithout recognizing th a t the study of the groups themselves and their representations necessarily has to precede the study of their invariants. Decisive for the developm ent of the theory of groups was the use E. Galois (1832) made of groups of perm utations for the investigation of algebraic equa­ tions; he recognized th a t the relation of an algebraic extension K to its ground field k is to a large extent determ ined by the group of automorphisms. His theory m ay be described as the algebraic relativity theory of a finite set of num bers which are given as the roots of an algebraic equation.4 Galois's brief allusions remained for a long tim e a book of seven seals. Only by C. Jordan's T raits des Substitutions (1870) was the newly gained field opened up to a wider circle of m athem aticians. The algebraic problems connected with the elliptic

29

VECTOR INVARIANTS

and m odular functions—partition, transform ation, complex m ultiplication— furnished the most im portant m aterial for the new concepts. Going ahead in this direction F. Klein and H. Poincare created the theory of autom orphic functions. While Galois’s theory deals with finite groups, infinite discrete groups here come to the fore. C rystallography became the motive for a detailed study of infinite discrete groups of m otions.5 S. Lie founded a general theory of continuous groups from the infinitesimal standpoint, and showed its im ­ portance by m any applications to geometric questions and differential equations.6 The theory of representations of groups by linear transformations was created above all by G. Frobenius7 during the years 1896-1903. Burnside, independent of him, and I. Schur in continuance of his work, found an essentially simpler approach by emphasizing the representing m atrix itself rather than its trace, the Frobenius character. For Lie’s infinitesimal groups E. C artan dem on­ strated the fundam ental propositions concerning structure and representations.8 The m atter is closely connected with hypercomplex num ber systems or algebras. After H am ilton’s foundation of the quaternion calculus (1843), and a long period of more or less fomal work in which the nam e of B. Peirce is outstanding, Molien (1892) was really the first to win some general and profound results in this direction.9 Of param ount im portance for the m odern development is W edderburn’s paper10 of the year 1908, where he investigates associative algebras in an arb itrary num ber field fc; also I. Schur’s study of irreducible representations in an arbitrary num ber field (1909) should be mentioned as fundam ental.11 Since then the development has been pushed ahead, in America chiefly by L. Dickson’s and A. A. A lbert’s efforts, in Germ any through E. N oether and R. Brauer. This brief enum eration of names m ust suffice here in place of a real history, as our link to the p a st.12 The bibliography will help to round out the picture w ith respect to m odern times.

2. The main propositions of the theory of invariants I t will be convenient, before going on, to illustrate the notion of vector invari­ an t (C hapter I, §5) by two fam iliar examples, the sym m etric group and the orthogonal group. In the theory of algebraic equations one is led to consider sym m etric functions f ( x i , x2 , • • • , x n) of n argum ents x x, • • • , x n , i.e. functions invariant under the group irn of all n! possible perm utations of the n arguments. These perm u­ tations are obviously linear transform ations of the n-dimensional vector x = ( x i , x2 , • * • , x n). The elem entary sym m etric functions

30

THE CLASSICAL GROUPS

whose roots are x \ , x2 , • • • , x n .

T hus

V>l(z) =

£

i

Xi ,

3'i%k )

(2 .2)