Tables of Dominant Weight Multiplicities for Representations of Simple Lie Algebras 0824772709, 9780824772703
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Table of contents :
Preface
Contents
1. Finite Dimensional Representations of Simple Lie Groups
2. Weight Algorithms
3. Biographical Table of Simple Lie Groups and the Positive Roots Table
Table 1. Biographical Table
A_n (n \geq 2)
B_n ( n \geq 3 )
C_n (n \geq 2)
D_n (n \geq 4)
E_6
E_7
E_8
F_4
G_2
Table 2. Positive Roots Table
A2, A3, A4, A5
A6, A7
A8
A9
A10
A11
A12
B2, B3, B4
B5, B6
B7
B8
B9
B10
B11
B12
C2, C3, C4
C5, C6
C7
C8
C9
C10
C11
C12
D4, D5, D6
D7
D8
D9
D10
D11
D12
E6
E7
E8
F4, G2
4. Multiplicity Tables
Table 3. Dominant Weight Multiplicity Table
A2
Class 0
Class 1
A3
Class 0
Class 1
Class 2
A4
Class 0
Class 1
Class 2
A5
Class 0
Class 1
Class 2
Class 3
A6
Class 0
Class 1
Class 2
Class 3
A7
Class 0
Class 1
Class 2
Class 3
Class 4
A8
Class 0
Class 1
Class 2
Class 3
Class 4
A9
Class 0
Class 1
Class 2
Class 3
Class 4
Class 5
A10
Class 0
Class 1
Class 2
Class 3
Class 4
Class 5
A11
Class 0
Class 1
Class 2
Class 3
Class 4
Class 5
Class 6
A12
Class 0
Class 1
Class 2
Class 3
Class 4
Class 5
Class 6
B2
Class 0
Class 1
B3
Class 0
Class 1
B4
Class 0
Class 1
B5
Class 0
Class 1
B6
Class 0
Class 1
B7
Class 0
Class 1
B8
Class 0
Class 1
B9
Class 0
Class 1
B10
Class 0
Class 1
B11
Class 0
Class 1
B12
Class 0
Class 1
C2
Class 0
Class 1
C3
Class 0
Class 1
C4
Class 0
Class 1
C5
Class 0
Class 1
C6
Class 0
Class 1
C7
Class 0
Class 1
C8
Class 0
Class 1
C9
Class 0
Class 1
C10
Class 0
Class 1
C11
Class 0
Class 1
C12
Class 0
Class 1
D4
Class 0
Class 1
D5
Class 0
Class 1
Class 2
D6
Class 0
Class 1
Class 2
D7
Class 0
Class 1
Class 2
D8
Class 0
Class 1
Class 2
D9
Class 0
Class 1
Class 2
D10
Class 0
Class 1
Class 2
D11
Class 0
Class 1
Class 2
D12
Class 0
Class 1
Class 2
E6
Class 0
Class 1
E7
Class 0
Class 1
E8
Class 0
F4
Class 0
G2
Class 0
References
Citation preview
PU RE
A N D
A P P -L I E D
MATHE M AT IC S
'
'
A Series of Monographs and Textbooks
TABLES OF DOMINANT WEIGHT MULTIPLICITIES FOR REPRESENTATIONS OF SIMPLE LIE ALGEBRAS
M. R. Bremner R.V Moody J. Patera
I
Tables of Dominant Weight Multiplicities for Representations of Simple Lie Algebras M. R. Bremner New Haven, Connecticut
R. V. Moody Department of Mathematics University of Saskatchewan Saskatoon, Saskatchewan, Canada
J.
Patera
Centre de Recherche de Mathematiques Appliquees Universite de Montreal Montreal, Quebec, Canada
MARCEL DEKKER, INC.
New York and Basel
PURE AND APPLIED MATHEMATICS A Program of Monographs, Textbooks, and Lecture Notes
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EDITORIAL BOARD M. S. Baouendi Purdue University
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Marvin Marcus University of California, Santa Barbara W. S. Massey Yale University Leopoldo Nachbin Centro Brasileiro de Pesquisas Fisicas and University of Rochester Anil Nerode Cornell University
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K. Yano, Integral Formulas in Riemannian Geometry (1970) (out of print) S. Kobayashi, Hyperbolic Manifolds and Holomorphic Mappings (1970) (out of print) V. S. Vladimirov, Equations of Mathematical Physics (A. Jeffrey, editor; A. Littlewood, translator) (1970) (out of print) B. N. Pshenichnyi, Necessary Conditions for an Extremum (L. Neustadt, translation editor; K. Makowski, translator) (197 I) L. Narici, E. Beckenstein, and G. Bachman, Functional Analysis and Valuation Theory ( 197 l) D.S. Passman, Infinite Group Rings (1971) L. Dornhoff, Group Representation Theory (in two parts). Part A: Ordinary Representation Theory. Part B: Modular Representation Theory (1971, 1972) W. Boothby and G. L. Weiss (eds.}, Symmetric Spaces: Short Courses Presented at Washington University (1972) Y. Matsushima, Differentiable Manifolds (E.T. Kobayashi, translator) (1972) . L. E. Ward, Jr., Topology: An Outline for a First Course (1972) (out of print} A. Babakhanian, Cohomological Methods in Group Theory (1972) R. Gilmer, Multiplicative Ideal Theory (1972) J. Yeh, Stochastic Processes and the Wiener Integral (1973) (out of print) J. Barros-Neto, Introduction to the Theory of Distributions (1973) (out of print} R. Larsen, Functional Analysis: An Introduction (1973) (out of print) K. Yano atid S. Ishihara, Tangent and Cotangent Bundles: Differential Geometry (1973) (out of print} C. Procesi, Rings with Polynomial Identities ( 197 3) R. Hermann, Geometry, Physics, and Systems (1973) N. R. Wallach, Harmonic Analysis on Homogeneous Spaces (1973) (out of print) J. Dieudonne, Introduction to the Theory of Formal Groups (1973) l Vaisman, Cohomology and Differential Forms (1973) B. -Y. Chen, Geometry of Submanifolds (1973) M. Marcus, Finite Dimensional Multilinear Algebra (in two parts) (1973, 1975) R. Larsen, Banach Algebras: An Introduction (1973) R. 0. Kujala and A. L. Vitter (eds.}, Value Distribution Theory: Part A; Part B: Deficit and Bezout Estimates by Wilhelm Stoll (1973) K. B. Stolarsky, Algebraic Numbers and Diophantine Approximation (1974) A. R. Magid, The Separable Galois Theory of Commutative Rings (1974) B. R. McDonald, Finite Rings with Identity (1974) J. Satake, Linear Algebra (S. Koh, T. A. Akiba, and S. Ihara, translators) (1975)
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Other Volumes in Preparation
Tables of Dominant Weight Multiplicities for Representations of Simple Lie Algebras
Library of Congress Cataloging in Publication Data
Bremner, M. R., [date] Tables of dominant weight multiplicities for representations of simple Lie algebras. (Monographs and textbooks in pure and applied mathematics ; v. 90) Bibliography: p. 1. Lie algebras-Tables. 2. Representations of algebras-Tables. 3. Lie groups-Tables. 4. Representations of groups-Tables. I. Moody, R. V., [date]. II. Patera, Jiri. III. Title. IV. Series. QA252.3.B74 1985 512'.55 84-19984 ISBN 0-8247-7270-9
COPYRIGHT© 1985 by MARCEL DEKKER, INC.
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Current printing (last digit): 10987654321
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Preface
The number of applications of the theory of simple and semisimple Lie groups both to mathematics and physics continues to grow with no end in sight. Although in recent years many of the new developments have been in the direction of infinite dimensional representations, the finite dimensional representations continue to be the basic tool in the majority of applied situations. Here there are a number of standard, but difficult, computational problems, among which we might mention the finding of suitable bases for representation spaces, the splitting of various symmetry classes of tensors, the computation of characters, character decomposition of class functions, and the determination of subgroup reductions. The first question to arise in almost any finite dimensional problem (in particular in those mentioned above) is the determination of the weights and the weight multiplicities of various representations of the group at hand. There are several documented [1-4] (and no doubt many undocumented) computer implementations of weight algorithms which attest to the fundamental nature of this problem. This book will, we hope, obviate the need for many of these programs, the main part of it being precisely tables of weights and weight multiplicities for the simple Lie groups of rank n for 2 < n < 12. This work is an early part of a more ambitious program of developing efficient and solidly based methods of general applicability for computing in compact Lie groups [5-9]. The algorithms need to be fast enough to compute useful information about the higher rank groups which are presently of interest to physicists and mathematicians, and general enough that they do not have to rely on special features of particular groups or series of groups. As a consequence there are two basic premises in this particular work: (1) all the simple Lie groups are placed on equal footing; (2) inherent symmetry is exploited as much as possible.
For anyone simply using the tables, (1) is not especially relevant. On the other hand, the effect of the second premise is immediately evident in the tables, where one sees that only dominant weights appear. Every other weight is linked to one of these by a sequence of Weyl reflections. Each weight on a given Weyl orbit has the same multiplicity and each orbit has a unique dominant representative. Restricting to dominant weights not only makes it possible to succinctly write down the relevant iii
iv
Preface
information but also represents an enormous increase in computational efficiency. In this respect the algorithm which we use (see Chapter 2) is a substantial improvement over any other in the literature. Basically it is a modified version of Freudenthal's classic formula but preceded by a direct computation of all the dominant weights of a given representation. Our implementation is in Pascal. Time was not an important factor in the computation (the table of 104 representations of £ 8 in this volume took about 18 minutes of computing time on a Cyber 835) and there is no difficulty either in extending the rank or the number of representations beyond what we have listed here. The volume contains three tables. The first two offer useful information about each simple Lie group and algebra. These are the Biographical and Positive Roots Tables. In one form or another this information can be found elsewhere in the literature. The main content of the volume is the original Dominant Weight Multiplicity Table. It is set out with one subtable for each congruence class (see Chapter 1) of each simple Lie group of rank n with 2 ,:;;; n ,,;; l 2. Within each subtable there are the "first" 52 representations of the given congruence class (note that the ordering is not by dimension!). In each of the cases G2 , F 4 , and E 8 , since there is only one congruence class, we have given the first 104 representations. Thus every group has at least 104 representations listed. Chapter 1 is a compendium of relevant facts about representations related to the tables of the volume and their use. No proofs and only a few references are given. The aim of the section is to systematize the facts for readers who may have experience in working with the representations but whose principal interests are elsewhere. A thorough explanation can be found, for instance, in Ref. 10. In Chapter 2 we describe the weight space and weight multiplicity algorithms. Chapter 3, the Biographical Table and the Positive Roots Table, contains basic biographical information about each type of Lie group (A, B, etc.). Included are facts about the roots, exponents, Cartan matrices, etc. To paraphrase a remark in the Introduction of Freudenthal and de Vries [11], it probably contains every possible fact except the one that is actually needed at the moment. The main part of the book is the Multiplicity Table of Chapter 4, preceded by brief instructions on how to use it. The development of the program constituted the major part of the Master's thesis of one of us (M.B.) at the Department of Computer Science, Concordia University, Montreal, where the computation also took place (Cyber 835). We would like to thank Cliff Grossner, Lily Lam, and Larry Thiel for their assistance with file transfers and the final printing of the tables, and Randy Funk for his help in early stages of the program development. It is also a pleasure to thank Wendy McKay, who assisted in formatting the Dominant Weight Multiplicity Tables and the Biographical Table. Two of us (R.M. and J.P.) would like to thank the University of Montreal, the California Institute of Technology, the University of Saskatchewan, and the Aspen Center for Physics for their hospitality during the work on the book. We also acknowledge the Special Collaborative Project Grant from the Natural Sciences and Engineering Research Council of Canada which was essential for successful completion of the project. M. R. Bremner R. V. Moody J. Patera
Contents
iii
Preface 1. Finite Dimensional Representations of Simple Lie Groups
1
2. Weight Algorithms
6
3. Biographical Table of Simple Lie Groups and the Positive Roots Table
8
65
4. Multiplicity Tables
340
References
V
1 Finite Dimeli.sional Representations of Simple Lie Groups
Let K be an arbitrary simply connected simple compact Lie group of rank n with Lie algebra k. Set S = 1, · · ·, The structure theory of K and its unitary representations proceeds through the choice of a maximal torus T, i.e., maximal connected abelian subgroup of K. Let g be the complexification k[ of k and let h C g be the Cartan subalgebra of g determined by T. Thus h = t[ , where t is the subalgebra of k corresponding to T, and ~: = is a real Euclidean space (under the Killing form) of dimension n. The exponential mapping determines an exact sequence
l
nl.
Pt
(1.1) The subgroup Q is the co-root lattice and is a discrete subgroup of hlR of rank n. Let rr : K-+ SU(V) be any (unitary) representation of K on a finite dimensional complex vector space V and let drr: k -+ End(V) be the corresponding representation of k. Complexification determines the representation (drr)[: g-+ End(V) of g. Then V decomposes into weight subspaces relative to T: V=
a:, /1.ED(V)
V('A.)
(1.2)
'
where D.(V) C h; (real dual space of hlR) is the weight system of V and the V('A.) are the corresponding weight subspaces of V. Thus V('A.)
= Iv
EV
I for all
x
E
hIR, rr(exp2rrix)v
= e 2rril\(x)v}.
(1.3)
We are primarily concerned with determining Q(V) and the corresponding weight multiplicities dim V('A.), 'A. E D.(V). The adjoint representation Ad: K -+ Aut (g) leads to the root space decomposition
g = h tll
tll
g(a:),
(1.4)
C is the weight lattice P. All weight systems D.(V) lie in P and Pis generated as a subgroup of by them. In particular, the Z·span Q of the roots lies in P. The index [P: Q] of Q in P, the index of connection, is the finite number
hi
(1.9)
[P: Q] = det(A).
The weight system D.(V) of any given irreducible representation (rr, V) of K lies en. tirely in one coset of P/Q. This results in the partitioning of the irreducible representa· tions of K into [P:Q] classes called congruence classes. In the construction of weight multiplicity tables it is always possible, and is always most efficient, to work within a single congruence class at any one time. An element A E P is called dominant if (A, ai) ;;;,, 0 for i E S, and the set of dominant weights is denoted by p++_ Then p++ = lN w
1
+ . . . + lN w n'
(1.10)
where w 1 , · · · , wn, the fundamental weights, are defined by
2(.wi, ai)/(ai, a) = Dij'
(1.11)
Here lN is the set of nonnegative integers and oij is the Kronecker function. Weights A of P are labeled by their coordinates A1 , · · · , /\n in terms of the basis w 1 , · · ·, wn. In every irreducible representation (7T, V) there is a unique weight subspace V(A) such that
1T(g(a)) V(A) = 0
for all a E 6.+.
(1.12)
This highest weight, I\ = Av, is dominant and its multiplicity is 1. The correspondence VI-+ Av is a bijective correspondence between irreducible representations (up to isomorphism) and elements of p++. This is Weyl's classification of irreducible unitary representations of K. We denote the irreducible representation with highest weight A E p++ by VA. The weight system D.(VA) of VA lies in the set (1.13) In the case of the adjoint representation the highest weight is the highest root (the unique root i cA for which ici is maximized). The lattice P is partially ordered by the level function L. Define f5 E hIR by (aJJ) = 1 for all i E. S. Then L : P -+ t if L(/\) > L(n. Specific expressions for the level vector 2/J = I = (1 1 , · · ·, ln)T are given in Table 1. Putting/\= (/\ 1 , /\ 2 , · · ·,\),one rewrites (1.14) as 11
L(f\) = 1 + 2,',, l.f\ .. l l
(1.15)
i=l
If we imagine that the weight system is partitioned into levels by the values of ~c· i I in (1.13), then in fact the number of levels is precisely L(/\). For each a e b. we may choose e±a e g(a) and h°' e hIR so that
[h°', e±a]
= ±2e±a'
[e°', e_°']
= h°'.
(1.16)
The h°' are uniquely determined by (1.16) and satisfy
f\(h°') = 2(/\,a)/(a,a)
for all/\ e
h;_.
(1.17)
The choice of e+_a: has one degree of freedom. The Lie subalgebra (Ce -0'. + Ce O'. + Ch O'. of g is isomorphic to sl(2,C). It is possible to choose the e+ uniquely to within sign -°' (Chevalley basis) so that under the action of sl(2, (C) on Vin any representation (IT, V) of g, (1.18)
RO'.:= exp e_°' exp(-e°') exp e -a acts as an element of order 4 and
IT(R ) V(/\) = V(f\ - f\(h )a) for all /\ e P. °' °' There is a resulting action on the weights
(1.19)
/\ f-+ /\ - f\(h °')a.
(1.20)
Define for each a e b.
by (1.21)
h;_
The operators r°' are genuine reflections (the Wey! reflections) of relative to the Euclidean geometry ( ·, . ) of the transposed Killing form. They generate a finite group W, the Wey! group of K (relative to T). Alternatively W arises as the quotient group N/T, where N is the normalizer of Tin K. According to (1.19) (1.22) Thus
(1) WS1(V) = S1(V) (2) for all weW and for all /\ES1(V) dim V(/\) = dim V(w/\).
(1.23)
This internal symmetry of the weight systems is an important ingredient in the theory of simple Lie groups. It is an indispensable ingredient in our approach to computational methods and theory [5, 7, 8, 9]. This will become apparent as we go on.
Chapter 1
4
The dominant chamber C in
C: =
l-P eh; I (sp, o);;, o, i
hm is defined by es\.
(1.24)
C is a closed convex cone the boundaries of which lie in the reflecting hyperplanes Hi:= jsp Eh; I (sp,o) = o} of the r/ = rOl.i' i e S. The group Wis generated by r 1, · · ·, rn. Each W-orbit J:1/t\, A Eh;, has exactly one point in C. In particular, each W-orbit of a weight system D.(V) has precisely one dominant weight. The entire orbit lies in D.( V) and the multiplicities are constant along the orbit. Of course, the highest weight heads an orbit of multiplicity one. The determination of the dominant weight t in an orbit J:1/t\, A E P, is a purely mechanical procedure. One writes the coordinates (A 1 , · · ·, An) of A relative to the basis of fundamental weights, i.e., A = "'i:,\wr Then A is dominant if and only if A.? 0 for each i. If some A.l < 0, then perform the reflection r.l on A. The level z L(riA) of r/ is greater than that of A. At most I~+ I reflections are required to bring A into p++. Conversely, given t e p++ the entire orbit Wt can be constructed by the following procedure [7] : We define a graph "'E,(n with nodes corresponding to distinct elements of Wt and edges "colored" by the index set S. Distinct A, A1 of Wt are joined by an i-edge (i e S) if and only if A1 = r/. Then L(n is connected. Define the depth of a node A to be the minimal number of edges in a path from t to A. Now L(n is created by induction on depth starting from the headnode For each node A of depth d, consider the coordinates (A 1 , · · ·, An) of A. For each i for which Ai > 0 there is an i-edge down from A to r? and the depth of r? is d + l. From the point of view of efficiency it is clear that one should restrict attention to dominant weights and their multiplicities, only going out of the dominant chamber when it is unavoidable. For example, consider the case of the representation
r
2
of D 8 whose highest weight is 0000000. Its dimension is 6435. There are 5 orbits; corresponding multiplicities and orbit sizes are as follows: Dominant weight
Multiplicity
Orbit size
2 0000000
1
128
0 0000010
1
1792
0 0001000
3
1120
0 0100000
10
112
0 0000000
35
1
This little table summarizes the essential information about the representation. The main body of the volume provides precisely this information about each representation. The reader can easily see how the multiplicity tables (Chapter 4) are used by looking under representation #10 in the D 8 class O table and also looking at the orbit size row, labeled O.S.
Finite Dimensional Representations
5
Determination of orbit sizes is a straightforward procedure. If/\ is the unique dominant weight of an orbit WI\ in P then define J = J(/\) C S in terms of the labels (/\ 1 , · · · , /\n) of /\ by J
= Ii E s
= 01.
I\
Define WJ to be t~e group generated by W and WI\ has precisely
[W: W] = J
~ IWJ I
(1.25)
jrj
lj
E
J}.
Then WJ is the stabilizer of/\ in
(1.26)
elements. Now the subgroups WJ of W (so-called parabolic subgroups) are again Weyl groups. In fact WJ is the Weyl group of the semisimple Lie algebra whose Dynkin diagram r J is obtained from the diagram r of K by deleting all nodes and incident edges which are not indexed from the subset J of S. The group WJ is the direct product of the Weyl groups of the connected components of rJ. For example, in D 8 0
for I\= 0001000,
rJ
is
0
0-0-0
I
0-0-0
(1.27)
and we have the decomposition WJ =" W(A 3 ) X W(D 4 ) into the Weyl groups of A 3 and D 4 with order I W1 I= 2 3 (4!)2. A list of the orders of the Weyl groups for the simple Lie algebras appears in Table 1. In the multiplicity tables (Chapter 4) the orbit sizes (1.26).are shown in the row labeled O.S.
2 Weight Algorithms
In this chapter we describe the algorithm by which the dominant weight multiplicities were computed. The mathematical development of this is found in [5], where one may also find a worked-out example. Given an irreducible representation V" with highest weight /\ = (/\ 1 , · · ·, /\n), the determination of its dominant weight multiplicities proceeds in two steps: (1) determination of the dominant weights; (2) determination of the multiplicities. (la). Inductively we construct subsets L 0 , L 1 , L 2, · · · of p++ as follows:
j/\}
Lo =
Lk = h,,eP++ - _U Li I "I= J 1/J -> Si:t6(Fz) -> 1 (exact) through the natural mapping of 1/J onto Sp(Q/2P,a) where a is the alternating form on Q/2P induced by (•,•)
16
Chapter 3
Table 1. (continued)
E8
0
Numberin (Bourbaki
Numbering (Dynkin)
Marks
Dimensions of fundamental representations
sum of positive roots
~;
ii
ii
4
5
5
146325270
220
5
4
6
6899079264
270
6
3 1
ll
6696000
182
2
3875
92
6
3
7
ll
Quadratic form matrix
Cartan matrix 0
0
0
0
0
0
2
3
4
6
4
2
3
2 -1
0
0
0
0
0
3
6
8 10 12
8
4
6
-1
0
0
0
0
4
8 12 15 18 12
6
9
2 -1
0
0
0
5
10 15 20 24 16
0 -1
6
12 18 24 30 20 10 15
2 -1 -1
0 -1
2
0
0 -1
0
0
0 -1
0
0
0
0 -1
0
0
0
0
0
0
0
0 -1
Index of connection:
2
-1
0
4
8 12 16 20 14
7
10
2
0
2
4
6
7
4
5
0
2
3
6
9
12 15 10
5
8
;;e1;;ej (Hi w -> O(q) -> 1 (exact) through the natural action of w on Q/20, where q is the quadratic form induced by (•, •)/2
17
Biographical Table and Positive Roots Table Table 1. (continued)
Numbering
Dimensions of fundamental representations
Marks
sum of positive roots
Level vector
52
22
16
1274
42
30
273
30
42
26
16
22
Quadratic form matrix
Cartan matrix 2
-1
0
0
2
3
2
1
-1
2
-2
0
3
6
4
2
0
-1
2
-1
2
4
3
3/2
0
-1
2
1
2
3/2
1
0
Inoex of connection: Roots:
1
!ei!ej (1.!.1
I»
S!: Cl)
N
\D
w
0
Table 2.
( continued)
ALGEBRA B9
---------It
II II It It
II It
II II II
It
II II II II II It It It It It
II II II It
II II II It
It
II II
It
II II II ti
1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24 25 26 27 28 29 30 31 32 33 34 35 36 37
OMEGA 0 1 0 1 -1 1 -1 0 1 1 0 -1 -1 1 -1 1 0 0 0 -1 0 -1 1 0 1 0 0 0 -1 1 -1 1 0 1 0 0 0 0 -1 0 -1 1 -1 1 0 1 0 0 0 0 -1 0 -1 1 -1 1 0 1 0 0 0 0 0 0 0 -1 0 -1 1 -1 1 0 1 0 0 1 0 0 0 0 0 0 0 -1 0 -1 1 -1 1 0 -1 1 0 1 0 0 0 0 0 0 0 0 0 -· 1 0 0 --1 1 0 -1 1
0 0 0 1 1 -· 1 1 -1 0 -1 0 0 0 0 0 0 1 0 0 0 -1 1 0 0 0 0 -1 1 0 0 0 0 0 -1 1 0 0
0 0 0 0 0 1 0 1 --1 1 -1 0 1 -1 0 0 --1 0 0 0 0 0 0 0 0 0 1 0 0 0 0 0 -1 1 0 0 0
0 0 0 0 0 0 0 0 1 0 1 -1 0 1 -1 0 1 -1 0 0 1 -1 0 0 0 0 -1 0 0 0 0 0 0 0 0 0 0
0 0 0 0 0 0 0 0 0 0 0 1 0 0 1 -1 0 1 -1 0 0 1 -1 0 0 0 1 -1 0 0 0 1 1 -1 0 0 0
0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 1 0 0 0 0 0 1 0 -1 2 0 0 0 0 1 0 -1 2 0 0 1 -2 0 0 1 0 -1 2 0 0 -2 1 -1 0 0 0 1 0 -··1 2 0 0 1 -2
ALPHA 1 2 1 1 0 1 1 1 0 1 1 1 0 0 0 1 1 1 0 0 0 1 1 1 0 0 0 0 0 1 1 1 0 0 0 0 0 1 1 1 0 0 0 0 0 0 0 1 1 1 1 1 0 0 0 0 0 0 0 1 0 1 1 1 0 0 0 0 0 0 0 0 0 0
2 2 2 1 1 1 1 1 1 1 1 1 0 1 1 1 0 1 1 1 0 0 1 1 1 1 0 0 1 1 1 1 0 0 0 1 1
2 2 2 2 2 1 2 1 1 1 1 1 1 1 1 1 1 1
1 1 0 1 1 1 1 1 0 1 1 1 1 1 0 0 1 1 1
2 2 2 2 2 2 2 2 1 2 1 1 2 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 0 1 1 1 1
2 2 2 2 2 2 2 2 2 2 2 1 2 2 1 1 2 1 1 1 2 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1
2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 1 2 2 1 1 2 2 1 1 1 1 2 1 1 1 1 1 2 1 1 1 1
2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 1 2 2 2 1 1 1 2 2 1 1 1 0 2 2 1 1 1
2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 1 0 2 2 2 1 0 0 2 2 2 1 0
EPSILON 1 0 1 1 0 1 0 1 1 0 1 0 0 0 1 0 1 0 0 0 1 0 1 0 1 0 0 0 1 0 0 1 0 1 0 0 0 0 0 0 0 1 0 0 1 0 1 0 0 0 0 0 0 1 0 0 1 1 0 0 0 0 0 0 0 0 1 0 0 0 0 1 1 0 0 1 0 0 0 0 0 0 0 0 0 0 1 0 0 1 0 1 0 0 1 0 0 0 0 0 0 0 0 0 0 0 0 1 1 0 0
0 0 0 1 1 0 1 0 0 0 0 0 1 0 0 0 1 0 0 0 0 1 0 0 0 0 0 1 0 0 0 0 0 0 1 0 0
0 0 0 0 0 1 0 1 0 1 0 0 1 0 0 0 0 0 0 0 1 0 0 0 0 0 1 0
o. 0 0 0 0 1 0 0 0
0 0 0 0 0 0 0 0 1 0 1 0 0 1 0 0 1 0 0 0 1 0 0 0 0 0 0 0 0 0 0 0 1 0 0 0 0
0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 1 0 0 0 0 0 0 0 0 1 0 0 0 1 0 0 0 0 1 0 0 0 1 1 0 0 0 0 0 0 1 0 0 0 1 1 0 0 0 0 0 0 0 -1 0 1 0 0 0 1 1 0 0 0 0 0 0 0 -1 0 0 -1 1 0 0 1 0 0 0 0 1 0 0 0 0 0 -1
n
:r Ill
"O M
..., (t)
w
II II II II II II II II II II II II II II II II II II II II II II II II II II II II II II II II II II 11 II II II II II II II II II
38 39 40 41 42 43 44 45 46 47 48 49 50 51 52 5:3 54 55 56 57 58 59 60 61 62 63 64 65 66 67 68 69 70 71 72 73 74 75 76 77 78 79 80 81
C) 0 -1 0 1 0 1 1 0 0 0 0 1 -1 0 -1 0 0 0 1 -1 0 -1 0 0 1 0 0 0 0 -1 1 0 0 0 0 -1 0 1 0 0 0 0 ·-1 1 0 0 0 1 -1 1 0 0 0 1 -1 1 1· -1 0 0 0 0 0 0 0 0 0 -1 0 0 0 -1 0 0 1 0 0 0 -1 0 1 0 0 0 -1 0 0 1 0 0 0 -1 0 1 0 0 1 0 -1 0 1 -1 1 0 -1 1 0 0 1 -1 0 1 0 0 1 -1 0 0 0 -1 0 0 0 0 1 0 0 -1 0 0 1 0 0 0 -·1 0 1 0 0 0 -1 0 0 1 ·o 1 0 0 -·1 1 0 1 -1 0 -1 0 1 1 -i 0 -1 1 0 1 -1 0 0 ·o 1 0 0 0 1 -1 O· 0 0 0 0 -1 0 0 0 -1 0 0 0 1 0 -1 0 0 0 0 1 0 -1 0 0 0 1 1 0 -1 0 1 0 1 -1 0 --1 0 1 1 -1 0 .o -1 1 1 -1 0 0 -1 1 1 -1 0 0 0 1 1 -1 0 0 0 0 0 0 0 0 0 0 -1 0 0 0 0 0 0 -1 0 0 -1 0 0 0 2 0 -1 0 0 0 2 -1 0 0 -1 2 -1 0 0 0 0 -1 2 -1 0 0 0 -1 2 -1 0 0 0 -1 2 ··1 0 0 0 0 2 -1 0 0 0 0 0 0 0 0 0 0 0 0
-1 0 0 0 1 0 -1 2 0 0 1 -2 -1 0 0 0 0 0 1 0 -1 2 0 0 1 -2 -1 0 0 0 0 0 ·o 0 -1 2 0 0 1 -2 -1 0 0 0 0 0 0 0 0 0 0 2 ·o 0 1 -2 -1 0 0 0 0 0 0 0 0 0 0 0 1 0
2 -2
-1 0 0 0 0 0 0 -1
0 0 0 0 0 0 0 2
0 1 0 0 0 0 0 0 1
0 0 0 0 0 0 0 1 0 0 0 0 0 0 0 1 0 0 0 0 0 0 0 0 1 0 0 0 0 0 0 0 0 1 0
1 1 0 0 0 0 0 1 1 0 0 0 0 0 0 1 1 0 0 0 0 0 0
1
1 0 0 0 0 0 0 0 1 1 0 0 0 0 0 0 0
1
0 0
1 1 0 0 0 0 1 1 1
0 0 0 0 0
1
1 1 0 0 0 ,0 0 1 1 1 0 0 0 0 0 0 1 1 0 0 0 0 0 0 0 1 0 0 0
1 1 0 0 1 1 1
1 1 0 0 0 0 1 1 1 1 0 0 0 0 1 1 1 0 0 0 0 0 0 1 1 O·
0 0 0 0 0 0 1 0 0 0 0
1 1 0 1 1 1 1 1 1 0 0 1 1 1 1 1 0 0 0 0 1 1
1 0 0 0 0 0 0 1 1 0 0 0 0 0 0 0 1
0 0 0 0 0
1 1 1 1 1 1 1 1 0 0 1
1 1 1 1 0 0 0
1
1 1 1 0 0 0 0 0 0 1 1 0 0 0 0 0 0 0 1 0 0 0 0 0 0
1 0 1 1 1 1 1 0 0 1 1
0 0 2 1 1 1 0 0 0 2 1
1 1 0 0 0 1 1 1 1 0 0 0 0 0 1
1 0 0 0 0 1 1 1 0 0 0 0 0 1 1 1 0 0 0 0 0 0 1 1 0 0 0 0 0 0 0 0
1
1
1 0 0 0 0 0 0 0 1 0 0 0 0 0 0 0
1
0 0 2 2 1 0 0 0 0 2 2 1 0 0 0 0 0 2 1 0 0 0 0 0 0 2 1 0 0 0 0 0 0 0 1 0
0 0 0 0 0 0 0 1
0 1 0 0 0 0 0 0 1 0 0 0 0 0 0 0 1
0 0 0 0 0 0 0 1 0 0 0 0 0 0 0 0 1 0 0 0 0 0 0 0 0 1 0
1 0 0 0 0 0 0 0 0 0 0 0 0 0 1 0 0 1 0 1 0 1 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 1 0 1 0 1 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 1 1 0 0 0 1 0 0 -1 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 1 0 0 1 0 -1 1 0 -1 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 1 0 0 1 -1 0 1 -1 -1 0 0 0 0 0
0 0 0 0 0 1 1 0 0 0 0 0 0 0 0 0 0 -1 0 0 0 1 1 0 1 0 0 0 0 0 0 -1 -1 0 0 0 0 1 1 0 0 1 0 0 0 -1 -1 0 0 0 0 0 0 0 0 0 0 1 0 1 0 -1 -1 0 0 0 0 0 0 0 0 0 0 0 0 1 1 -1 -1 0 0 0 0 0 0 0 0 0
0 -1 0 -1 0 0 0 0 1 0 0 1 0 0 0 0 -1 0 0 -1 0 -1 0 0 0 0 0 1 1 0 1 0 0 0 0 0 0 -1 0 0 -1 0 -1 0 0 0 0 0 0 0 0 1 0 1 0 0 0 0 -1 0 0 -1 0 -1 0 0 0 0 0 0 0 0 0 0 0 0 1 1 0 0 1 0 -1 1 0 0 -1 -1 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 1 0 0 1 -1 -1 1 0 -1 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 1 0 0
ex,
o·
..,
OQ
PJ
"c=;·
~
~ -{ PJ 0~ PJ :l C.
-a
0
;:;: "'
J ::::i
c.. -0
.,,0
;:;:
:.:· Cl)
;ilJ 0 0
.,,....
-l l>J
CT
J
"C
:::,
c=;· ~
..., l>J
O"
iii l>J
::::,
c.. "C
~0
!:!'.
< ('!)
;;;, 0
....0
...,"'
ALGEBRA C6
l 2 3
,;....
OQ
ONEGA 2 0 0 0 l 0 -2 2 0 l -1 l -1 0 l 0 -1 l 0 -2 2 -1 l -1 0 l 0 0 -1 0 -1 0 l l 0 0 0 -2 0 0 -1 l -1 l 0 0 0 l 0 -1 0 0 -1 l -1 l 0 l 0 0 0 0 0 0 -1 0 0 -1 l -1 l 0 l l 0 0 0 0 0 -1 0 0 -1 l -1 l l l -1 l 0 0 0 0 0 0 0 -1 0 0 -1 2 -1 2 -1 2 -1 0
0
0
0 0 0 l 0 l -1 l -1 0 2 -1 0 0 0 0 0 l -2 l 0 l -1 -1 l l -1 0 0 -1 2 -1 0 0
0 0 0 0 0 0 0 0 l 0 l -1 0 l -1 l l -1 l -1 2 -1 l -1 0 0 l -1 0 0 -2 2 -1 0 0 0
0
0
0 0 0 0 0 0 0 0 0 l 0 0 l -1 0 l -1 0 0 l -1 0 0 l -1 0 0 0 2 -1 0 0 0 0
ALPHA 2 2 l 2 0 2 l l 0 l l l 0 0 0 l l l 0 0 0 l l l 0 0 0 0 0 l l l 0 0 0 0 0 l l l 0 0 0 0 0 0 0 l l l 0 0 0 0 0 0 0 l l l 0 0 0 0 0 0 0 0 0 l l 0
2 2 2 2 2 2 2 2 2 2 2 l 2 2 l l 2 l l 0 2 l l 0 0 l l 0 0 0 0 l 0
0
0 0
l l l l l l l l l l l l l l l 0 l l 0 0 l l 0 0 0 l 0 0 0 0 l 0 0 0 0 0
EPSILON o. 0 0 0 2 0 0 0 0 l l 0 0 0 2 0 0 0 0 0 0 l l 0 0 0 l l 0 0 0 l 0 0 l 0 0 0 0 2 0 0 0 0 0 0 l l l 0 l 0 0 0 0 0 0 l l 0 0 0 l 0 0 l l l 0 0 0 0 0 0 0 0 2 0 0 0 l 0 l 0 l 0 l 0 0 0 0 -1 l 0 0 0 0 0 0 0 l l l 0 l 0 0 0 0 -1 0 l 0 0 0 0 -1 l 0 0 0 0 0 2 0 0 l 0 l 0 0 0 0 -1 0 0 l 0 0 0 -1 0 l 0 0 -1 0 l 0 0 0 l 0 0 0 l 0 -1 l 0 0 0 0 -1 0 0 0 l 0 0 -1 l 0 0 0 -1 0 l 0 0 2 0 0 0 0 0 l -1 0 0 0 0 -1 0 l 0 0 0 l -1 0 0 0 0 l -1 0 0 0 0 l -1 0 0 0 0
l>J
O"
iii
w
1.0
Table 2.
(continued)
ALGEBRA C7
---------1
z 3 4
s
6
7
8 9
10 11
1Z 13
14
15 16 17
18 19
zo Z1 zz
Z3
II II II II II II II II II II II II II II II II II II II II II II II
Z4 ZS Z6 Z7 Z8 Z9 30 31 3Z 33 34 35 36 37 38 39 40 41 4Z 43 44 45 46 47 48 49
011EGA
z 0 1 -z z
0 0 0 1 -1 1 -1 0 1 0 -1 1 0 -z z -1 1 -1 0 1 0 0 -1 0 -1 1 0 1 0 0 0 0 -z 0 -1 1 -1 1 0 1 0 0 0 -1 0 0 -1 1 -1 1 0 1 0 0 0 0 0 0 -1 0 o· -1 1 -1 1 0 1 0 0 0 0 0 0 -1 0 0 -1 1 -1 1 0 1 0 0 0 0 0 0 0 0 0 -1 0 0 -1 1 -1 1 0 1 0 1· 0 0 0 0 0 0 0 -1 0 0 -1 1 -1 1 1 1 -1 1 0 0 0 0 0 0 0 0 0 0 -1 0 0 -1 Z -1 Z -1 Z -1 0 0
0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 1 0 0 0 0 0 0 0 1 0 0 0 -1 1 0 0 1 o, 0 0 -1 1 0 0 0 -1 1 0 z 0 0 0 -1 1 0 0 0 -1 1 0 0 -1 0 1 0 1 0 0 0 -1 0 1 0 -1 0 1 1 -1 0 0 -z z 0 0 1 -1 1 0 0 -1 0 1 1 -1 0 0 0 1 -1 0 -1 0. 1 0 1 0 -1 1 0 1 -1 0 1 -1 0 0 1 -1 0 0 0 -z z 0 -1 1 -1 1 1 -1 1 0 1 -1 0 0 1 -1 0 0 -1 0 0 0 0 -1 0 1 -1 1 -1 1 1 -1 1 0 1 -1 0 0 -1 0 0 0 0 0 0 0 0 0 -z z Z -1 0 -1 -1 Z -1 0 Z -1 0 0 -1 0 0 0 0 0 0 0 0 0 0 0
ALPHA
z z z z z 1 z z z z 0 z z z z 1 1 z z z z z z 0 1 1 1 0 z1 zz zz 0 0 1 z z 1 1 1 1 1 z 0 0 1 z z 0 1 1 1 z 1 1 1 1 1 0 0 0 z z 0 0 1 1 z
0 1 0 0 0 1 0 0 0 0 1 0 0 0 0 1 0 0 0 0 0 1 0 0 0 0 0 1 0 0 0 0 0 0 1
1 1 0 0 1 1 0 0 0 1 1 0 0 0 1 1 0 0 0 0 1 1 0 0 0 0 1 1 0 0 0 0 0 1 0
1 1 0 1 1 1 0 0 1 1 1 0 0 1 1 1 0 0 0 1 1 1 0 0 0 1 1 0 0 0 0 0 1 0 0
1 1 1 1 1 1 0 1 1 1 1 0 1 1 1 1 0 0 1 1 1 0 0 0 1 1 0 0 0 0 0 1 0 0 0
z z z z z zz z z zz zz z 1 z 1 1 z z 1 z 1 1
1
1 z z 1 z 1 1 1 1
1
1 1 1 0 0 1 1 1 0 0 0 1 1 0 0 0 0 0 1 0 0 0 0
1 0
z
1 1 0 0
z
1 1 0 0 0 1 1 0 0 0 0 0 1 0 0 0 0 0
1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 0 1 1 1 0 0 1 1 0 0 0 1 1 0 0 0 0 1 0 0 0 0 0 1 0 0 0 0 0 0
EPSILON
z
1 0 1 0 1 0 0
0 1
z
0 1 0 0 1 0 0
0 0 0 1 1 0
z
0 0 0 0 0 1 0 1 0 1 0 0
0 1 0 0 1 0 1 0 0 1 0 0 0 0 z 0 0 1 0 0 1 0 0 1 0 0 0 0 0 0 1 0 0 1 0 0 1 0 0 1 0 0 0 0 0 0 0 0 0 0 1 0 0 1 0 0 1 0 0 1 0 0 0 0 0 0 0 0 0 0 1 0 0 1 0 0 1 0 0 0 1 0 0 0 0 0 0 0 0 0 0 0 0 0 1 0 0 1 0 0 0 1 0 0 -1 1 0 0 0 0 0 0 0 0 0 0 0 0 1 0 0 0 1 0 1 0 -1 () 1 0 -1 0 0 0 0 0 0 0 0 0 0 0 0 1 0 0 0 1 -1 0 0 1 -1 0 0 1 -1 0 0
0 0 0 0 0 0 0 0 1 0 1 0 0 1 0 0 1 0 0 0
z
0 0 0 0 1 0 0 0 -1 0 1 0 0 -1 0 0 1 0 -1 0 0 0 0 1 -1 0 0 0
0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 1 0 0 0 0 0 1 0 1 0 0 0 1 0 0 1 0 -1 0 0 0 1 1 0 0 -1 -1 0 1 0 1 0 0 -1 -1 0 0 0 z 0 0 1 0 -1 -1 0 0 0 0 0 1 1 0 -1 -1 0 0 0 0 0 0 0 z 0 1 -1 -1 0 0 0 0 0 0 0 0 0
(")
::r l>J
....
"C
(1) ..,
J
CT
co" l>J :i
c.. "'tl
0
V>
;:;:
:.:· Cl)
;;c 0 0
Iii° -I l>J
CT
co"
.i,:..
1.0
Vo 0
Table 2.
(continued)
ALGEBRA D4
---------II 1 II 2 II 3 II 4 II 5 II 6 II 7 II B II 9 II 10 II 11 II 12
OMEGA 1 0 0 1· 1 -1 -1 0 1 -1 0 1 0 1 1 -1 1 -1 -1 1 1 1 -1 1 0 -1 0 -1 2 -1 0 -1 2 2 -1 0
ALPHA 1 2 1 1 0 1 1 1 1 1 0 1 0 1 1 1 0 0 0 1 0 0 1 0
0 1 1 1 -1 1 -1 -1 2
-1
0 0
1 1 1 0 1 0 1 0 0 0 1 0
EPSILON 1 1 0 0 1 0 1 0 0 1 1 0 1 0 0 1 0 -1 1 0 0 1 0 1 0 -1 0 1 0 -1 1 0 0 0 1 1 0 1 -1 0 0 0 1 -1 1 -1 0 0
1 1 1
1 0
1
0 0 1
0 0 0
ALGEBRA D5
----------
II II II II II II II II II II II II II II II II II II II II
1 2 3 4 5 6 7 B
9 10 11 12 13 14
15
16
17 18 19 20
OMEGA 0 1 0 1 -1 1 0 1 0 -1 1 -1 1 -1 0 1 0 1 0 0 0 -1 0 0 -1 1 -1 1 0 1 0 1 0 -1 1 0 -1 1 -1 1 1 -1 1 1 0 -1 0 0 -1 2 0 -1 0 -1 2 -1 2 -1 0 -1
ALPHA 1 2 2 1 2 1 2 1 1 1 0 1 1 1 1 1 1 1 1 0 0 1 0 1 1 0 1 1 1 1 1 0 0 1 0 0 1 0 1 1 1 1 0 0 0 0 0 0 1 0 0 0 0 1 0 1 0 0
0 0 0 1 1 -1
0 0 0 1 1 1 1 -1 1 1 -1 1 1 -1 -1 -1 -1 1 1 -1 -1 -1 0 0 0 2 -1 -1 2 0 0 0 0 0
1 0
1 1 1 1 1 0 1 1 0 1
0 0 1 0 0 0 0
EPSILON 0 0 0 1 1 1 0 0 1 0 1 0 0 0 1 0 1 0 1 0 0 0 1 0 1 0 0 1 0 1 0 -1 0 1 0 0 0 1 1 0 0 0 0 1 1 0 -1 0 1 0 0 -1 1 0 0 0 0 1 0 1 0 -1 0 0 1 0 -1 0 0 1 0 -1 0 0 1 1 0 0 1 0 0 0 1 -1 0 0 1 -1 0 0 0 1 -1 0 0 0 0 0 1 -1
1 1 1 1 1 1 0 1 1 0 0 1
0 0 0
1
0 0
1
0 0 0 0
2 2 2 1 1 1 1 1 1 1 1 1 1 1
2 2 2 2 2 1 2 1 1 1 1 1 1 1
ALGEBRA D6
---------II II II II II II II II II II II II II II
1
2 3
4
s 6
7 B 9 10 11 12 13 14
OMEGA 0 1 0 0 0 0 1 0 0 0 1 -1 -1 0 1 0 0 0 0 -1 1 1 0 0 -1 1 -1 1 0 0 -1 0 1 0 1 1 0 -1 0 1 0 0 -1 0 -1 1 1 1 0 -1 0 0 1 1 0 0 0 1 1 -1 0 -1 1 -1 1 1 -1 1 0 0 -1 1 -1 1 0 0 1 -1 -1 1 0 0 1
-~
ALPHA 1 2 1 1 0 1 1 1 0 1 1
1
0 0 1 1 0 0 0 1
0 1 1 1 0 1 1 1
1
1 1
1 1
1 1 1 0 1 1 0 1 0
1 1 1 1 1 1 1
1 1 0 1 1 0 0
EPSILON 1 1 0 1 0 1 0 1 1 1 0 0 0 1 0 1 0 0 0 0 1 0 1 0 1 0 0 1 0 0 0 0 1 0 1 0 0 1 0 1 0 0
0 0 0
1 1
0
0 0 0 0 0
1
1 0 0 1 0 0 0 0 0 1 0 0 0 0 0 -1
0 0 0 0 0 0 0 0 1 -1 0 1 -1 0
n
:r
l>J "Cl
~ -,
w
II II II II II II
15 16 17 18 19 20
II 21 II II II II II II II II II
22 23 24 25 26 27 28 29 30
0 0 0 -1 1 0 0 0 -1 1 0 0 0 0 -1 2
0 -1 -1 1 0 0 0 -1 1 1 0 0 0 -1 2 -1
-1 1 1 0 1 -1 -1 1 1 -1 0 -1 0 2 -1 0
0 0 0 1 -1 1 1 1 -1 0 -1 2 -1
1 -1 1 -1 0 -1 1 -1 0 0 0 -1 2 -1 0 0 0 0 0
1 1 -1 -1 0 1 -1 -1 0 0 2 -1 0 0 0 0
0 0 0 0 1 0 0 0 0 1 0 0 0 0 0 1
0 0
·o 1 1 0 0 0 1 1 0 0 0 0 1 0
0 1 1 1 1 0 0 1 1 0 0 0 0 1 0 0
1 1 1 1 0 1 1 1 0 0 0 1 0 0 0 0
1 0 1 0 0 0 1 0 0 0 0 0
1
o· 0 0
0 0 0 1 1 0 1 0 0 1 0 0 0 0 -1 0 0 1 0 0 1 0 0 -1 0 0 0 1 0 -1 0 1 0 0 0 1 0 0 0 1 0 -1 0 0 0 1 0 -1 0 0 1 0 -1 0 0 1 0 -1 0 0 0 0 1 1 0 0 0 0 1 -1 0 0 0 0 0 0 1 -1 0 0 0 0 1 -1 0 0 0 1 -1 0 0 0 0 1 -1 0
1 1 0 0 0 1 0 0 0 0 1 0 0 0 0 0
OMEGA 1 2 3 4 5 6 7 8 9 10 11 12 13
14
15 16 17 18 19 20 21 22 23 24 25 26 27 28 29 30 31 32 33 34 35 36 37 38 39
40 41
42
0
..., s,,
-a ::,
o·
~
-I s,,
2:: (1) s,,
:::l 0..
ALGEBRA D7
---------II II II II II II II II II II II II II II II II II II II II II II II II II II II II II II II II II II II II II II II II II II
~ (IQ
0 1 -1 1 -1 1 0 -1 1 0 -1 1 1 0 0 -1 -1 1 0 0 0
-1
1 0 0 0 0
-1
1
0 0 0 0 -1 1 0 0 0 0 0 -1 2
1 -1 0 0 1 0 -1 1 0 -1 1 0 0 0 -1 1 1 0 0 -1 -1 1 0 0 0 0 -1 1 0 0 0 0 -1 1 1 0 0 0 0 -1 2 -1
ALPHA 0 1 1 -1 -1 0 0 0 0 1 0 0 0 -1 1 0 0 0 -1 1 1 0 0 0 -1 -1 1 0 1 0 0 -1 1 1 -1 0 0 0 -1 2 -1 0
0 0 0 1 1 -1 1 -1 0 -1 0 0 0 0 0 0 0 0 1 0 0 0 1 -1 1 1 0 1 -1 -1 -1 1 1 -1 0 0 -1 0 2 -1 0 0
0 0 0 0 0 1 0 1 -1 1 -1 0 0 1 -1 0 0 1 -1 0 0 1 -1 0 0 0 1 -1 0
1
1 1 -1 0 0 -1 2
-1
-1 0 0 0
0 0 0 0 0 0 0 0 1 0 1
-1
0 0 0 0 0 0 0 0
1
0 1 1
1 -1 0 0 1 1 -1 1 1 -1 -1 -1 1 1 -1 1
1 -1
-1 -1 0 0 1 1 1 1 -1 -1 -1 0 0 0 0 -1 1 1 -1 -1 -1 0 0 0 0 0 0 0 2 -1 -1 2 0 0 0 0 0 0 0 0 0 -1
1 1 0 1 0 1 0 0 1 0 0 1 1 0 0 0 0 1 0 0 0 0 1
0 0 0 0 0 1 0 0 0 0 0 1 0 0 0 0 0 0 1
2
1
1 1 1 1 0 1 1 0 1 1 1 0 0 1 1 1 0 0 0 1 1
0 0 0 0 1 1 0 0 0 0 1
1
0 0 0 0 0 1 0
2 2 2 1 1
1
1 1 1 1 1 1 1 0 1 1
1
1 0 1
1
1 1 0 0 0 1 1 1 0 0 0
1 1
0 0 0 0 0 1 0 0
2 2 2 2 2 1 2 1 1 1 1 1 1 1
1
1 1 1 1 1 1 1
1 0 1 1 1 1 0 0 0 1 1 0 0 0 0 0 1 0 0 0
·2 2 2 2 2 2 2 2 1 2 1 1 1 2 1 1 1 1 1 1 1 1 0 1 1 1 1 0 0 1 1 1
0 0 0 0 1 0 0 0 0 0
EPSILON 1 1 1 1 1 1 1 1 1 1 1 0 1 1 1 0 1 0 1 0 1 0 0 1 0 1 0 0 0 0 1 0 0 0 0 0 0 1 0 0 0 0
1 1 1 1 1 1 1 1 1 1 1 1 0 1 1 1 0 0
1
1 0 0 0 1
1
0 0 0 0 1 0 0 0 0 0 1 0 0 0 0 0 0
1
1
1
0 1 0 1 0 0 1 0 0 1 0 0 0 0 1 1 0 0 0 0 1 0 0 0 0 0
0 1 0 1 0 0 1 0 0 1 1 0 0 0 0 1 0 0 0 0 1 0 0 0 0 1 0 0 1 0 0 0 0 0 0 0 0 0 1 0 1 0 0 0 0 0 0 0 0 0 0 0 1 1 -1
0 0 0 0 0 0 1 0 0 0 0 0 0 0 1 0 0 1 0 0 0 1 0 0 0 0 0 0 1 0 0 1 1 0 0 0 0 0 1 0 1 0 0 0 0 0 1 0 1 0 0 0 0 1 0 1 0 0 0 0 0 -1 0 0 0 0 0 1 1 0 0 1 0 0 1 0 0 1 0 0 0 0 0 0 -1 -1 0 0 0 0 1 0 1 0 0 0 0 0 1 1 1 0 0 0 -1 -1 0 0 0 0 0 0 0 0 -1 0 1 0 0 1 1 0 1 0 0 -1 0 1 0 0 0 1 0 0 -1 0 -1 0 0 0 0 0 -1 0 0 1 0 0 1 0 0 0 1 0 -1 0 0 1 0 -1 0 0 1 0 -1 0 0 0 0 -1 0 -1 0 0 0 1 0 0 0 1 0 0 0 1 -1 0 0 0 1 -1 0 1 -1 0 0 0 0 0 1 -1 -1 0 0 0 0 0 0 0 0 0
""O
0
V,
;:;:
:;::· (1)
;;o 0
0 ..... V,
-I s,,
2:: (1)
Table 2.
(continued)
ALGEBRA DB
---------II II II II II II II II II II II II II II II
II II II II II
II
II II II II II II II II
II
II II II II II
II
II II II II II II II II II II II II II
II
II
It II II II II
1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24 25 26 27 28 29 30 31 32 33 34 35 36 37 38 39 40 41 42 43 44 45 46 47 48 49 50 51 52 53 54 55 56
OMEGA
0 1 1 -1 -1 0 1 0 -1 1 1 0 0 -1 -1 1 1 0 0 -1 -1 1 1 0 0 0 0 -1 -1 1 1 0 1 0 0 0 0 -1 -1 1 -1 1 1 0 0 0 0 0 0 -1 0 -1 -1 1 1 0 0 0 0 0 0 0 0 -1 -1 1 1 0 0 0 0 0 0 0 0 0 0 -1 -1 1 1 0 0 0 0 0 0 0 0 0 0 -1 -1 1 1 1 0 0 0 0 0 0 0 0 0 0
0 0 0 1 0 0 1 0 0 -1 1 0 -1 1 0 1 0 -1 0 1 0 0 -1 1 0 0 -1 1 -1 1 0 0 -1 0 0 0 -1 0 1 1 0 -1 0 0 0 0 0 0 0 0 0 -1 1 -1 1 0 0 0 0 0 0 0 0 0 0 0 0 -1 0 -1 1 0 1 0 0 1 0 0 0 0 0 0 0 ,1 0 -1 1 -1 1 0 -1 1 0 1 0 0 0 0 1 0 1 -1 0 0 -1 -1 0 1 0 -1 1 -1 1 0 1 0 1 0 1 -1 1 -1 0 0 0 -1 0 0 -1 0 -1 1 -1 1 1 1 1 -1 1 -1 0 -1 0 0 0 0 0 0 0 -1 0 0 0 0 -1 2 -1 2 -1 0 -1 2 -1 0 -1 2 -1 0 0 2 -1 0 0 0
0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 1 0 0 0 0 0 1 0 0 -1 1 1 0 0 0 1 0 0 -1 1 1 0 -1 1 0 1 -1 1 0 0 -1 1 1 0 -1 1 0 1 -1 1 -1 -1 1 0 0 -1 1 1 0 -1 1 0 1 -1 1 -1 -1 -1 0 0 -1 1 1 0 -1 1 0 1 -1 -1 -1 1 -1 0 0 0 0 0 0 1 1 -1 0 1 0 1 -1 1 -1 -1 -1 0 0 0 0 0 0 0 0 1 -1 1 1 1 -1 1 -1 -1 -1 0 0 0 0 0 0 0 0 0 0 0 -1 0 2 2 -1 -1 -1 0 2 -1 0 0 0 0 0 0 0 0 0 0 0 0 0 0
ALPHA
1 1 0 1 0 1 0 0 1 0 0 1 0 0 0 1 1 0 0 0 0 1 0 0 0 0 0 1 0 0 0 0 0 1 0 0 0 0 0 0 1 0 0
0 0 0 0 1 0 0 0 0 0 0 0 1
2 1 1 1 1 1 0 1
1
0 1 1 0 0 1 1
1 0 0 1
1
1 0 0 0 0 1 1 0 0 0 0 1 1 0 0 0 0 0
1 1 0 0 0 0 0 1 1 0 0 0 0 0 0 1 0
2 2 2 1 1 1 1 1 1 1
1 1 0 1 1 1 1 0
1
1 1 1 0 0 1 1 1 1 0 0 0 1 1 1 0 0 0 0 1 1
1 0
0 0 0 1
1 0 0 0 0 0
0 1 0 0
2 2 2 2 2 1 2 1 1 1 1 1 1 1 1 1 1 1 1
1
1 1 0 1 1 1 1 1 0
1 1
1 1 1 0 0 0 1 1 1 0 0 0 0 1 1 0 0 0 0 0
0 1 0 0 0
2 2 2 2 2 2 2 2 1 2 1 1 2 1 1 1
1 1
1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 0 0 1
1
1 1 0 0 0 0 1
1
0 0 0 0 0
0
1 0 0
0
0
2 2 2 2 2 2 2 2 2 2 2 1 2 2 1 1 1 2 1 1 1 1 2 1
1 1 1 0 1 1 1
1
0
0
1 1
1 1 0 0 0
1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 0 1 1 1 0 1 0 1 1 0 1 0 0 1 0 1 0 0 0 1 0 1 0 0 0 0
1
0 1
0
0 0 0 0 0 0
1 1 0 0
0 0 1 0 0 0 0 0 0
0 1
0 0 0 0 0
1
1 1 1 1 1 1 1 1
1
1
1 1 1 1 1 0 1 1 1 0 0 1 1 1
0 0 0 1 1 0 0 0 0 1 1 0 0 0 0 0 1
0 0 0 0 0 0 1 0 0 0 0 0 0 0
EPSILON
1 1 0 1 0 1 0 0 1 0 0 1 0 0 0 1 1 0 0 0 0 1 0 0 0 0 0 1 0 0 0 0 0 1 0 0 0 0 0 0 1 0 0 0 0 0
0 1 0 0 0 0 0 0 0 1
1 0 1 0 1 0 0 1
0 0 1 0 0 0 1 0 0 0 0 1 1
0 0 0 0 0 1 0 0 0 0 0
1
0 0 0 0 0 0 1 0 0 0 0 0 0 1 0 0 0 0 0 0 0 1 -1
0 0 0 0 0 0 1 0 0 1 0 0 0 0 1 0 0 1 0 0 0 1 0 0 1 1 0 0 1 0 0 0 0 0 0 1 1 0 1 0 0 1 0 0 0 0 0 0 0 1 0 1 0 1 0 1 0 0 0 0 0 0 0 0 0 0 0 0 0 1 0 1 0 0 1 0 0 0 0 0 0 0 0 0 0 0 0 0 1 0 0 1 0 0 0 1 0 0 1 0 1 0 0 0 0 0 0 0 0 -1 0 0 1 0 0 0 0 0 1 0 0 0 1 0 1 0 0 0 0 -1 0 0 0 0 -1 0 0 0 1 0 0 0 1 0 0 1 0 0 0 0 1 0 0 -1 1 0 0 0 -1 0 0 0 0 -1 0 0 0 1 0 0 0 1 1 0 0 0 0 1 0 -1 1 0 -1 0 0 0 0 -1 -1 0 0 0 0 0 0 0 C) 0 0 1 0 0 0 0 1 -1 0 0 0 0 1 -1 0 0 1 -1 -1 0 0 0 0 0 0 0
0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 1 0 0 0 0 0 1 0 1 0 0 -1 0 0 1 0 0 1 0 -1 -1 0 0 0 1 0 0 1 0 -1 -1 0 0 0 1 0 1 0 0 -1 -1 0 0 0 0 0 1 0 1 0 0 -1 -1 0 0 0 0 0 0 0 1 0 0 -1 -1 0 0 0 0 0 0 0 0 0 1 1 -1 0 1 -1 0 0 0 0 0 0 0 0 0 0
n:::, Pl "O
....
....
(1)
w
ALGEBRA D9
----------
II II II II II II II II II II II II II II II II II II II II II II II II II II II II II II II II II II II II II II II II II
1 2 3 4
5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24 25 26 27 28 29 30 31 32 33 34 35 36 37 38 39 40 41 II 42 II 43 II 44 II 45 II 46 II 47 II 48 II 49 II 50 II 51 II 52 II 53 II 54 II 55 II 56 57 II 58 II 59 II 60 II 61 II 62 II 63 II 64 II 65 II 66 II 67 II 68 II 69 II 70 II 71
"
II 72
a:,
OMEGA 0 1 0 1 -1 1 -1 0 1 1 0 -1 -1 1 -1 0 1 0 0 -1 0 -1 1 0 1 0 0 0 -1 1 -1 1 0 1 0 0 0 0 -1 0 -1 1 -1 1 0 0 1 0 0 0 -1 0 -1 1 -1 1 0 1 0 0 1 0 0 0 0 0 0 0 -1 0 -1 1 -1 1 0 -1 1 0 1 0 0 0 0 0 0 0 -1 0 -1 1 0 -1 1 -1 1 0 1 0 0 0 0 0 0 0 0 0 0 -1 -1 0 0 0 -1 1 -1 1 0 1 0 0 0 0 0 0 0 0 0 0 0 0 -1 0 0 -1 1 -1 1 0 1 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 -1 -1 0 1 -1 1 0 1 0 1 0 0 0 0 0 0 0 0 0 0 0 0 0 0 -1 0 -1 1 -1 1 1 1 1 -1 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 -1 0 -1 2 -1 2 -1 2 -1 0
0 0 0 1 1 -1 1 -1 0 -1 0 0 0 0 0 0 1 0 0 0 0 -1 1 0 0 0 0 -1 1 0 0 0 0 0 -1 1 1 0 0 0 0 -1 -1 1 0 0 1 0 0 0 -1 1 0 1 -1 0 0 0 -1 1 1 -1 0
0 0 0 0 0 1 0 1 -1 1 -1 0 1 -1 0 0 -1 0 0 0 0 0 0 0 0 0 0 1 0 0 0 0 0 -1 1 0 0 0 0 1 -1 1 1 0 0 1 -1 0 -1 -1 1 0 1 -1 0 0 0 -1 1 1 -1 0 0
0
0
0 0 0 0 0 -1 -1 2 -1 2 -1 0 0 0
0
0
0 0 0 0 0 0 0 0 1 0 1 -1 0 1 -1 0 1 -1 0 0 0 1 -1 0 0 0 0 -1 0 0 0 0 1 0 0 0 0 0 1 -1 1 0 0 0 1 -1 0 -1 1 1 0 1 -1 0 0 -1 -1 1 1 -1 0 0 0 0 -1 0 2 -1 0 0 0
0
0 0 0 0 0 0 0 0 0 0 0 1 0 0 1 -1 0 1 -1 0 0 0 1 -1 0 0 1 1 -1 0 0 1 -1 1 -1 0 0 1 -1 0 -1 0 0 1 -1 0 0 0 0 0 1 -1 0 0 0 1 1 1 -1 0 0 0 0 -1 2 -1 -1 0 0 0 0
0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 1 0 0 1 -1 1 0 0 1 -1 1 -1 0 1 -1 1 -1 0 0 1 -1 1 -1 0 0 1 -1 1 -1 0 0 0 1 -1 1 -1 0 0 0 0 -1 1 -1 0 0 0 0 0
0
-1 2 0 0 0 0 0 0 0
0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 1 0 0 1 1 -1 0 0 1 1 -1 -1 0 1 1 -1 -1 0 0 1 1 -1 -1 0 0 1 1 -1 -1 0 0 0 1 1 -1 -1 0 0 0 0 1 -1 -1 0 0 0 0 0 2 -1 0 0 0 0 0 0 0
ALPHA 1 2 1 1 0 1 1 1 0 1 1 1 0 0 0 1 1 1 0 0 0 1 1 1 0 0 0 0 0 1 1 1 0 0 0 0 0 1 1 1 1 1 0 0 0 0 0 0 0 1 0 1 1 1 0 0 0 0 0 0 0 0 0 1 1 1 0 0 0 0 0 0 0 0 0 0 0 1 1 1 0 0 0 0 0 0 0 0 0 0 0 1 1 1 0 0 0 0 0 0 0 0 0 0 0 0 0 1 1 1 0 0 0 0 0 0 0 0 0 0 0 0 0 1 1 1 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 1 1 0
2 2 2 1 1 1 1 1 1 1 i 1 0 1 1 1 0 1 1 1 1 0 0 1 1 1 1 0 0 1 1 1 1 0 0 0 0 1 1 1 0 0 0 0 1 1 1 0 0 0 0 0 1 1 1 0 0 0 0 0 1 1 0 0 0 0 0 0 0 1 0 0
2 2 2 2 2 1 2 1 1 1 1 1 1 1 1 1 1 1 1 1 1 0 1 1 1 1 1 0 1 1 1 1 1 0 0 1 1 1 1 1 0 0 0 1 1 1 1 0 0 0 0 1 1 1 0 0 0 0 0 1 1 0 0 0 0 0 0 0 1 0 0 0
2 2 2 2 2 2 2 2 1 2 1 1 2
1
1 1 1 1 1 1 1 1 1 1
1
1 1 1 1 1 1 1 1 0 1 1 1 1 1 1 0 1
1
1 1 1 0 0 0 0 1 1 1 0 0 0 0 0 1 1 0 0 0 0 0 0 0 1 0 0
0 0
2 2 2 2 2 2 2 2 2 2 2 1 2 2 1 1 2 1 1 1 1 2 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 0 1 1 1 1 1 0 0 0 1 1 1 1 0 0 0 0 0 1 1 0 0 0 0 0
0
0 1 0 0 0 0 0
2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 1 2 2 1 1 1 2 2 1 1 1 1 2 1 1 1 1 0 2 1 1 1 1 0 0 1 1 1 1 0 0 0 1 1 1 1 0 0 0 0 1 1 1 0 0 0 0 0
1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 0 1 1 1 1 0 1 0 1 1 0 1 0 0 1
0
0 0
1
0 1 0 0 0 1 0 1 0 0 0 0 1 0 1 0 0 0 0 0 0 1 0 0 0 0 0 0
1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 0 1 1 1 1 0 0 1 1 1 0 0 0 1 1 1 0 0 0 0 1 1 0 0 0 0 0 1 1 0 0
0
1 0 0 0 0 0
0 0 0 0 1 0 0 0 0 0 0 0 1 0 0 0 0 0 0 0
0
0
0
1 0 0 0 0 0
EPSILON 0 1 1 0 0 1 0 1 1 0 0 1 0 0 1 1 0 1 0 1 1 0 0 0 1 0 0 1 0 0 0 1 1 0 0 0 1 0 0 0 0 0 0 1 1 0 0 0 0 0 0 1 0 0 1 0 1 0 0 0 1 0 0 0 0 0 0 1 1 0 0 0 0 0 0 1 0 0 0 1 0 0 0 1 0 0 0 0 0 0 0' 1 0 0 1 0 0 0 0 1 1 0 0 0 0 0 0 1 0 0 0 0 0 0 0 1 0 0 1 0 0 1 0 0 0 0 0 1 0 1 0 0 0 0 0 0 0 0 0 0 0 0 0 1 0 0 0 1 0 0 1 0 0 1 0 0 0 1 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 1 1 0 0 0 1 0 0 0 1 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 1 0 0 1 0 0 0 1 0 0 -1 1 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 1 0 0 1 0 0 -1 0 1 0 -1 0 1 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 1 0 0 1 -1 0 0 1 -1 -1 1 0 0
0 0 0 0 0 1 0 1 0 1 0 0 1 0 0 0 0 0 0 0 0 1 0 0 0 0 0 1 0 0 0 0 0 0 1 0 0 0 0 0 0 1 1 0 0 0 -1 0 0 0 1 0 0 -1 0 0 0 0 1 0 -1 0 0
0 0 0 0 0 0 0 0 1 0 1 0 0 1 0 0 1 0 0 0 0 1 0 0 0 0 0 0 0 0 0 0 0 1 0 0 0 0 0 -1 1 0 0 0 0 -1 0 0 1 1 0 0 -1 0 0 0 0 1 0 -1 0 0
0
0 0 0 0 0 0 0 0 0 0 0 1 0 0 1 0 0 1 0 0 0 0 1 0 0 0 0 1 0 0 0 0 -1 1 0 0 0 0 -1 0 0 0 0 0 -1 0 0 1 0 0 0 -1 0 0 0 1 1 0 -1 0 0 0
0
0 0 0 0 0 1 0 0 0 0 1 -1 1 -1 0 -1 0 0 0 0 0 0 0 0 0 0 0
0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 1 0 0 1 0 0 0 0 1 0 0 -1 0 1 0 0 -1 0 0 1 0 0 -1 0 0 1 0 0 -1 0 0 0 1 0 0 -1 0 0 0 0 0 0 -1 0 0 0 0 0 1 -1 1 0 0 0 0 0 0
0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 1 -1 0 0 0 1 -1 0 0 0 1 -1 0 0 0 0 1 -1 0 0 0 0 1 -1 0 0 0 0 0 1 -1 0 0 0 0 0 1 -1 0 0 0 0 0
o·
OQ
-,
I»
,:J
:::;
0
~
-I
I» 0(1)
I» ~
c.. "'O
0
~.....
J
6
8 2
4
4
4
4
12
4
4 2
4 3
4
4
3
11
4
4
2 4
15
8
4
"O
(ti
16
-,
11
.i:,..
1
2
2 1
1
4 2
1
1
1 1
2
3
2
2
1
4
3
1
1
2
1
1
2
2 2
3
2 4
.1 1
2
1
1
4
2
2
2
u u
4
~
C
tt
n u n
~
~
~
e;· ~
~
~ ~
ro ~
~
n 2
1
4
~
2 2
1
4
~
3
2
1
4
4 2
2
'
'2
2 3
~
2
~ ~
2
1
n ll
n
1
M ~ ~ ~
~
n ® ~
~ ~
~ ~
.1
~
u
• ~
~ ~
00 ~
Table 3.
(continued) \,0
0 Al6EBRA Ali, REPRESENTATIONS Of CLASS 0
14 28 42 42 42 56 56 56 TO 84 TO 84 84 84 TO '8 ,a ,a '8126 140 84126 126 140 154 -98 '8154,210'112 112 112 112 U2,210 2'4126 126 154126 126 154 294126 168 140 140 140 168 154 S.P.
1 n tt 1 42 210 105105140 42 420 42G'U1
t
33 34 35
1
36
31
1
38 3'
40 41 42 43 44
45
1 1
4li
41 48
"' 5Q
51
..... "°
Table 3.
\.0
(continued)
N
ALGEBRA A6, REPRESENTATIONS Of CLASS 1 6 20 34 34 48 48 48
16
62
62
62 90 90
l 105 42 210
35 140 420 105 210 210 420 21
l
25
11 19
23
25 21
29
29
2, 31
31
,0
16 104 90 90
16 146 104 132 90
,0 104 216 118 104 104 118 160 146 104 146 118 132 114 202 118 132 118 146 114 160 216 132 132 132 132 114 286 146
1 42 105 210 42 105 420 260 42 105 140 420 420 420 31
31
33 33 35
35
35
35
1 1 l 210 420 840 105 21 210 21,0 210 840 210 42 105 630 140 260 210 42 420
31 31 31 31 31 31 39 39 39 39 41 41 41 41 41 41 43 43
43
43 43
43 43
43
1 1 42 630 630 21,0 260 210 45 45 45 45
45
42 420
45 41 41
S .P.
0 .s. LEVEL
,o
2 1 2 2 4 1 2 2 4 2 5 9 16 2 8 5 15 16 13 14 21 33 13 5 18 51 19 44 26 12 10 56 23 83 2, 15 30 13 11 112 115 40 9 88 l 140 189 588 490 840 016 515 352 800 410 116 156 100 704 646 292 450 800 310 820 115 680 800 860 924 700 168 264 475 292 480 450 600 352 460 936 395 700 520 160 106 288 240 860 442 552 266 500 320 009 200 Dln'N
0 0 0
4
10 4
0
0
3 0
0 2
9 10 11 12
5
24 15 16
4
30
30
10 13
8
21
13 14 15 16 11 18 19
10
4 0 0 0 1 1
20 21
22
23
24 25
26 21
28
29 30 31 32
33
2
1 0
2 3 0
0
1
0
0
0
0
3
4
0
39 40 41 42
43
44 45 46
70 120 105 19 15 1SQ 50 225 54 56
60
15 144 216 21,1 270
34 35 36 37
38
45
,0
80 60
40 66 144 55
40 155
21
42
4 16 10 32
3'
30
23
38
64 25
24 84 30 12
15
15
10 10 10
20
10
15 11
44 15
4 20 40 30 20 30 10 10 40 15 70
18
4 18
14 12
13 22
24
4 14 45
1 12
4
1
l
12 5
4
3
10
8 24
42
4
60
3 11 22
10 11 10
21
6
10 35
4
81 32
4 51
36 30
36
39
24
36 27
51
21
6
1 24 33 51
39
12
18 18 12
21,
12 18 18
21
16
15 15 24
18
3 20
27
10
13
12 14
12
13
4
18
4
10
12
l
4
10
1
4
35 28
4 31
12 18
13
1 16
93
"
24
4 19
17
10 11
10
12
4 13 14
12 4
NUfl8ER
6 155
54 63 75
24
24 18
6 16
4
49 50 51
50
1 12
24
4
48
75 10
10 36 18
4
41
'6
21
30 15
2
10 10 3,
24
10 4
21
44 18 66 26
15
17
4
YEI6HT
45 13 10 82 30 122 37 21 40 11 87 120 144 153 44
12
12
4
5
12
0
0
0
21
4
4 2
0
0
11
20 60
1 0
0
0
3
26
30
4 0
0
12
5 4
0 1 1 1 0 1
3 0 0
4
4
6
3
4
6
4
3
4
4
4
6
13
3 11
5 18 10
14
15 16
5
4
4 4
u
4
4
4
~
n a u ~
4
3
~ C ~ ~
n
0
~
~
~
n
~
~
~
~
~
~
ro
~
n
3 3
3
~ ~ ~
1
n
1 1
2
• D
~
a
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~
®
a ~ ~ ~ ~
~ ~ ~ ~
~ ~
~
w
Table 3.
( continued)
~
..i,..
AliEBRA Ai, REPRESEITATIOMS OF CLASS 2
-------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------10 24 24 38 38 21
" " 52
52 94
""
" 150 80 80 80 122 80 94 108 94 108 136 108 108
1 140 105 210 42 105 42 630 42 140 210 420
1 420 105 420 105 840 210 35 420 105
Mmmrnmmm™rnrnmmmmmrnm™rnrnrnrnmrururn 1
1
11200mmm0mm~m~m~~00~0
1 42 210 210 420 840 260 210 42 210 420
S.P.
o.s.
11 13 1' 23 23 25 25 21 29 29 31 31 31 31 33 33 35 35 35 35 3l 31 31 31 31 39 39 39 41 41 41 41 41 41 43 43 43 43 43 43 43 43 43 45 45 45 45 45 45 41 41 41 LEPEL 3
21
1
2
3
6
4
4
9
3 14
9
4 15
1
5 12 20 36 16 23 24 31 58 30 14 11 23 12 51 44 55 12 48
6 41 50 15 86 101 58 1' 40 19
~mfflmmm~~m~~-~rnrua~~~mm~mmwmru~m™m~m~mmm~~mm•~ffl~mM~~~m
Olft'N
----------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------0 1 0 0 0 0
0 0 1 0 0 1
0 0 0 1 1 0
1 1 0 0 0 1
3 0 0 0 0 1
4
5
0 0 0 0 2 1
1 0 1 0 1 0
1
8
1 0 0 2 0 0
0 2 0 0 1 0
1 0 1
0 0 0 0 0 5
0 1 1 1 0 0
2 1 0 0 1 0
4 0 0 0 1 0
1 0 0 1 1 1
2 1
0 0 3 0 0 0
1 0
4
0 0 0 0 2
1 0
1 2 0 1 0 0
0 1 1 0 1 1
1 0 0 0 2 2
1 1 2 0 0 0
3 1 0 1
0 1
0 1 1 0 0 3
•
1 0 0 0 1 4
• 3 1 0 0 0
3 0 2 0 0 0
5 0
0 1 0 1 2 0
0 0 2 1 0 1
2 0 0 2 0 1
1 2 0 0 1 1
2 0 1 0 0 3
1 0
2 2 1 0 0 0
0 0 2 0 2 0
0 1 0 1 1 2
2 0
1 4 0 0 0 0
4 1 1 0 0 0
• • • ' ---------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------6
0 0 2
0
0 0 2
0
0 3 0
0
1 0 3
•
0
2 0 1
0
1 0
0 0
0
1 2
UEI6HT
9 10 11 12 13 14 15 16 11 18 1' 20 21 22 23 24 25 26 21 28 29 30 31 32 33 34 35 36 31 38 39 40 41 42 43 44 45 46 41 48 49 50 51 NUftBER
---------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------4
10
24
4 15 20 30 10 10 1 12
34 25 36 15 56 45 11 50 20 21 40 56 115 39 50 55 a, 165 10 25 2, 40 20 105 90 120 180 120 24 15 16 15 16 30 12
4
3
11
5 30 15
6 24 13
16
3
12
10
3 13
5
5
2
4
5
4
5
4
•
3 4 4
21 10 36 55
30
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3 6
5 15
15
13 14
12 18 18 15
5 10
5 10
4 10
5
'
13 18
5 10
5
2
6
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4
1 21 28 38 64 52 40 10 24 38
15
11 44
4
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10 20
8
23 15 1 14 25
15 18
10
6 16 11 10
11 13
11
12 12 22 14
13
12
8 12 4
4
20
4 30
16 16 22 24 31 14
12 16
15
16 30 16 54 22
' 6
4
1l 41
6 21 30 52 32 15 22
4 16 16 18 22 12
5 10
6
2
15
,o
36 52 86 84 123 60 13 28 12
10 20 10 21 11 1' 11 24
•
4
45 50 "144 85 90 14 40
,o
4 39 28 42 34 24
4 10 10
4
12
"
84
4 12 1' 10 21 21 32 56 30
12
3
2
11 21 22 42 20
18 21
60 90 141 184 195 130 1, 45 115
1' 30 20 45 45 60 120 60
40
12 12 26 1' 16 12 30 36 18 15 10 14
4 2
,.
24 21 51 21 28 21 51 81 42 20 11 23 10 '3 51
6
4
30 45 39
12
9
11 12
6 18
13
6
6
5
4
5
4 4
2
2
3
6
4
8
4
4
1
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16
6
4 11 10
3
4
4
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n
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15
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16
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3
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Table 3.
(continued)
ALGEBRA 116, REPRESENTATIONS OF CLASS 3
12 16 54 16
40
54 40
21
41 410
96
54 54 68 110 68
68
81
82
82
82 110 96
96
96 96 166 124 96 110 110 190 124 110 110 138 124 138 138 138 138 114 194 138 138 138 151 151 138 114 151 180 166 194 166 S .P.
1 JS
42
1 105
13
n" "n n
H
1 105 410 110 41 420 630 110 210 630 420 110 140 140 420 420 H
n
H
H
41 410 160 210 840 105 110 420 840 110 630 105 210 410 105 410
2 41 420 14ll 420 105 420 420 520 110 410 21Q 410 105 0 .S.
n n n n n n n n n n n n nu u u a a a a a a au u u u u u u u u
45
45
45
45
45
45
~
~
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LML
1 1 J 1 1 4 1 5 1 11 11 6 8 10 14 11 3 11 35 10 43 8 18 33 51 14 31 10 15 40 26 42 11 43 18 41 31 58 60 169 61 38 69 53 31 35 112 84 110 196 318 313 110 260 014 450 386 104 560 600 056 901 025 468 131 500 100 248 014 248 280 190 008 316 900 600 144 192 340 580 200 425 950 336 466 120 815 040 152 111 480 185 136 808 JOO 900 044 DIM'N
UE!GHT
9 10 11 11
13
14 15 16 11
18
19 20
21
22
23
14 25
26 11
28
29
30
31
32 33
34 35
36
31
38
39
40
41
41
u
44 45
46
41
48
49
50
51
NUMBlR
--- -------- ------ ----- ---- -- --- - ---------------- ----- ----- --------------------------- - -- -- - - - --------------------------------- - ---------- - ----------------- -- --- -- --- - -- -- - -- ------------- -- -- - -- --- -- - - - - - -- --12
18
15
24
10
33
11 24 45
15
41
10 24 10
20
23
4 11
60
44 96
60
96
15
39
15 30 30
31
46
34 40
40
56
15
15 40
56
14
15 16
15 40
11
20
10 20
40
4
10 10 12 4
13 14 11 18 18 10
IS 10
11
16 11 14 10
16 54 11 48 4 30
10
4
14
15 11 11
1 11 11 14 18
4 4
12
76 128 60 12 45
34 44 14 51
30 20
16 30 26
10 11
18
3 11
14
10
10 16 10
19
24 10 14
10 16
21 18 54 100
39
21
12 38 35
so
11
54
81
65
45
10
60
55
26
30
45
15
44
51
33
19
55
34 30
15
19
35
11
11
4 13
45
90 96 326 105
29
66
so
190 39
30
30
36
55
46
56
31
60
26
45
60 168
10 20
25 16
40
15
30
16
11
40
15
11
30
15 30
20
40
40
34
15 14
13
14
11
16 14 17
33
31
96
21
16
10
10
15
24
11
14 16 54
10
15 10 44 19
11
10 30 45
10 10 10 30 15
78
20
30
1 11
10
11
13
11
11
15
40
10 15
11
18
13
80
11
11
10
11
12
11
17
21
15
4 11
4
12
11
11
4
22
J
11
11
14 12
10 24
14
13
14
10
4
10
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Table 3.
(continued)
\0
00
Al6EBRA Al, REPRESENTATIONS Of CLASS 0 0 16 32
48
48 64 48 64 64 64 '6 '6 80 80 80 '6 '6 112 112 112 112 112 112 144 128 160 '6 128 160 144 112 112 112 144 192 ll6 128 192 128 128 ll6 208 240 128 144 128 116 144 208 176 240 256 S .P.
11 1 56 420 168 168 56 560 TO 120 1ZO 280 280 840 840 680 420 420 336 336 168 840 168 840 1 15 25
Zl
2l 2, 31 33
35
35 31 37 1, 3'
39 41 41 41
41
5 111 56 168 280 040 168 280 336 680 120 680 336
43 43 43 43 43 45 45 45 45 45 4l 4l 4l 4T 4T
33 3 3 28 336 420 28 360 360 336 56 168 360 840 360 840 840
56 840 168 56 O.s .
4, 4, 4, 49 45 49 49 51 51 51 51 51 51 51 SI 51 SI 13 LEm
121ll55WW24tst5171lts~ts~l3251l'6251l~92TI~~13~H13mm~n2l~~llirn~nrn2lM 63 120 945 ,45 232 352 164 680 680 TI5 TI5 TOO 100 255 120 120 920 ,20 816 100 816 100 104 812 325 228 812 325 580 400 616 400 580 860 688 625 HO 605 605 688 264 021 544 320 544 100 320 264 100 on 320 Dlh'N
I 0 0 0 0 0 1
0
0
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1 · 0 0 0 0 2
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0 2 0 1 0 0 0
9 10 II 12 13 14 15 16
6
- ----------- ----------------------------------20
1
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24 24 15 15 30
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UE16HT
IT 18 19 20 21 22 23 24 25 26 2T 28 2, 30 31 32 33 34 35 36 31 38 39 40 41 42 43 44 45 46 41 48 4, so 51 NUhBfR
10 115 56 56 112 112 42 90 42 90
84 56 35 342 56 35 10 280 1'6 280 10 14 64 245 14 483 483 64 28 21 448 420 448 210 420 28 210 21 20
30 65 26 26 42 42 21 45 21 45 28 30 20 153 30 20 40 130 95 130 40
13 13
20 10
4
0
---------------------------------------------------------------------------------------- -------------------- ----------------------------------- ··-- .. - - -- -- ·- -- . -- -. - -----
14 64 64 35 35 TO
4
0 0 0 0 2 1 0
10
12 12 12 12 II 6
12
6
21
11 21
16 10 66 16
10 23 56 41 56
1 10 10 24
15
12
ts
30 15 20
6
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25
10
15
16
4
4 14 16
4
'8
98 24 13 10 120 80 120 45 80
68
34
36
55 56 45 45
34 68 20
56
45 36 15
24 24
16
30 16 15 30
10 10
11
12 12
32
24 12 12
4
4
6 32
6
30 15 11 4
4
18
4
10
4 4
4
15 6
1 15 10 4 15
11
5
13
12 4
16
4
10 10
15
4 16
4 2
45
15
10 11
15 10
15
12
4
45
11 18 30 15 15
24 II
12 2
55
13
62 21 62 15 21 ·
4 36 36 14
4
6
1, 105 15
9 228 228 40 19 15 212 195 232 105 m
24 46
15
4 13 21 25 21 13
4
40 110
4 20 25
15 20 15 30
24 10 10
10
4
23
10
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3
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n
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Table 3.
(continued)
0 0
Alb£BRA Al, REPRESENTATIONS OF CLASS 1 7 23
39
8 168
1 13 56 560 280 180 840 168 336 840 56 680 168 280 180 560 168 120 360 336 840 280 840
8
n n
39
28
55 55 55 81
~
32
32 N
71
N
11 103 11 81 103
¾ ¾
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81 81 103 103
40 40 40 42
81 11' 103 151 103 151 119 161 135 119 103 119 135 183 135 119 161 151 241 151 161 135 119 135 199 343 183 183 231 135 135 135 151 183 S .P.
42
56 840 168
31 1 53 56 840 360 680 168 56 840 680 336 336 56 336 840 840 040 360 56
42 44 44 44 44 44 44 %
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Table 3.
(continued) Vl
0
AL6EBRA Al 1, REPRESENTATIONS Of CLASS 4 32 56 BO 104 116 ~6 80 BO 104 104 128 104 152 128 200 128 104 z,6 116 128 152 116 128 200 200 152 152 128 152 224 212 116 116 248 152 152 116 320 2'6 224 200 200 224 3'14 3,2 152 248 248 248 416 116 116 S .P. I 15 83 3 21 5 2 83 5 1 23 110 55 I 1 15 I 21 166 41 I 2 33 11 23 211 5 5 166 138 5 15 13 5 5 5 1 1 4,5 660 66 132 12 5'14 840 860 5'14 ,40 ,40 920 320 495 320 840 160 132 960 120 ,40 no 160 660 940 920 160 880 440 660 980 980 840 320 120 no 520 66 320 910 264 880 160 132 660 200 660 940 9)
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u
4 2
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Table 3.
( continued)
ALGEBRA A12, REPRESENTATIONS or CLASS 6 -----------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------
0\ 00
42 68 68 94 120 120 94 120 146 120 198 94 198 224 146 302 ll2 198 432 198 120 224 120 146 216 302 146 146 328 432 146 ll2 588 112 198 224 146 224 ll2 ll2 224 ll2 250 198 302 198 328 !lZ 224 354 198 198 S .P, I 6 10 4 2 12 6 1 2 l2 8 2 102 1 180 51 25 l2 102 11 6 8 25 540 8 205 180 51 216 11 11 I 12 8 360 38 1 15 102 rummmm~rn•ru~~m~rnmrn~~u~•ru™~rnm~mrnrn•muam~~~mmfflm~mrummmmru~•o~. 43 53 55 ~
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61
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61 61 61 61 69 11 11
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n
15 11 11
11 19 19 19 19 H 83 83 83 85
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85 85 Bl 81 8l
89 89 89 89
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4 1 2 2 l 3 8 8 5 9 6 12 3 IQ 3 15 11 4 9 19 1n~nHS0~%mm~m~~mn™m~m•mmw~mrnmmmmmso~m™ >
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14 11 18 15 15 33 17
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6 35 36 43 14 44 75
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24 a, 11 10 56 48 66 11 46 56 56
26 51 18 18 65 53
46 36 42 53 33 48 102 54 30 45 98 118 42 10
26 42 18 14 53 49
34 2, 40 4, 2, 35 ,5 41 21 45 '2 104 39
42 37 52 62 31 42 120 60 24 55 113 12, 48
10 10 11 14
30 14
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45 56 61 42 56 128 68 34 55 120 146 52 10
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Table 3.
(continued)
AL6EBRA 83, REPRESENTATIONS Of CLASS 1
11 1, 21 21 35 51 43 51 59 s, 83 n
24 24 24 13 11
15 15 ,1 ea 123 "
48 24 24 24 48 24 24 24
u u n
~
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n
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" 101 101 131 m 123 111 m 13' 131 141 141 139 n, 163 155 1ss 221 111 1n 111 m
48 48 48 24 24 24 24 48 24 48 48 24 24 48 24 24
31 31 31
n
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n n n n » »
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43 43 43 43 43
m 1,s m 181 235 203 203 1,5 211 m
s .P.
48 48 48 24 48 48 24 48 24 24 24 24 48 48 24 48 48 48
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9 10 11 12 13 14 15 16 11 18 1' 20 21 22 23 24 25 26 21
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28 29 30 31 32 33 34 35 36 31
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46 47 48 49 50 51 NUflBER
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Table 3.
....-..J
( continued)
00
Al6EBRR 84, REPRESENTATIONS OF CLASS 0
16 12 16 20 36 24 32 28 40 64 36 44 40 52 48 48 68 52 100 56 64 12 12 60 80 68 16 104 16 12 144 84 80 84 100 108 88 88 116 96 112 92 148 104 108 100 104 196 128 S .P.
4 24
32 16 48
96 96 96 48 32 64 48 64
96 24 64 48
15 11 19 21 23 25 21
8 192 16 24 96 192 48 96 64 48 96 192
29 29 31 33 33 35 35 31 31 31 39 39 41 41
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24
96 192 48 96 64 192 48 192 32 192 96
o.s.
41 43 43 43 45 45 45 41 41 41 49 49 49 49 51 51 51 51 53 53 53 53 55 55 55 55 55 57 ST LDEL
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4 43 46 29 22 25 46 89 21 13 49 90 16113 32139 TO 4 26 28 1T 16 16 29 51 20 48
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41 33 25 22 24 30 63 27 41 34 18 16 TO 30 120 51
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Table 3.
.....
(continued)
00
0 ALGEBRA B4, REPRESEITRTIUIS OF CLASS 1
4 12 20 28 28 36 36 52 44 52 52 60 84 60 68 68 16 16 16 92 84 124 84 100 100 100 92 108 100 108 132 108 108 112 116 116 124 132 140 124 124 148 132 148 132 180 140 148 140 148 228 164 S.P, 16 64 96 64 64 16 192 64 192 96 64 192 64 192 1'2 96 192 64 64 192 64 64 384 16 96 192 192 192 192 64 192 192 1'2 64 384 '6 64 192 192 1,2 192 1,2 192 64 192 192 384 64 1'2 96 64 96 0 .s . 11 19 25 21 29 31 33 35 31 39 39
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43 43
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51 51 51 53 53 53 55 "
55 51 51 51 59 59 59 59 61 61 61 61 63 63
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63 65 65 65
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91011121314151611181'20212223242526212829303132333435363138394041424344~464148495051NUnBER
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Table 3.
(continued)
Al&EBRll 85, REPRESEIITRTIUIIS OF CLASS 0
4
8 16 12 20 1' 36 20 24 32 40 28 '4 32 36 44 52 40 '8 48 48 44 100 52 5' 52 12 12 56 80 60 64104 '8 16 64 '8 144 1' 80 80 84 12 100 108 80 16 116 88 H 84 S.P.
10 40 10 81 80 80 10 32 240 40 80 320 10 160 240 240 80 480 80 80 320 320 10 160 480 320 40 240 320 80 960 80 80 .240 320 640 1'0 10 240 160 32 480 960 80 240 480 '60 80 240 11 19 21 25 29 29 31 31 35 31 39 39 41 41 43 45 41 41 49 49 49 49 51 51 53 53 55 ss 55 51 51 51 59 59 59 59 59 11 55 65
0 0 0 0
1 0 0 0
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2 0 0 0
m 0
0
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''° 320
n 61 61 61 63 u 65 ,5 65 65 n n n n
0 .s . LEPEL
I I 2 3 4 5 1 1 11 1 1 15 11 2 22 33 23 13 2' 31 28 18 23 21 51 58 111 41 1 10 85 28 131 118 12 81 115 289 91 162 333 188 429 330 215 4'2 430 144 02s 003 935 290 005 129 tso 583 2n 8'5 400 1'0 111 215 033 595 ,so 520 152 1'8 650 595 945 915 344 915 1,0 001 010 085 314 445 150 930 s10 115 s1s 960 162 234 760 Din'N
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Table 3.
(continued)
AL6EBRA 85, REPRESENTIITIONS OF CLASS 1
5 13 21 29 29 31 31 53 45 45 53 61 53 85 61 61 69 11 69 93 11 11 11 125 85 85 85 101 101 93 109 93 tot 133 101 109 101 109 113 109 111 125 111 109 133 141 111 tn 14' 125 125 125 S .P. 1 1 1 1 1 32 160 320 160 320 640 160 160 32 960 320 640 6411 160 160 960 960 640 960 640 320 640 32Q 160 160 920 640 320 960 320 640 920 160 640 960 640 640 160 16' 960 160 32 920 920 640 960 960 960 640 960 920 640 0 .s. 16 26 N
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1 1 2211221241 1 1 3 10 5 1 4 28 24 4S 4S 22 36 91 134 141 160 151 128 219 131 64 183 513 292 264 480 214 510 034 251 4S1 896 198 901 302 160 140 612 151 288 114 351 425 168 013 151 358 212 '21 32 320 408 160 520 240 280 040 224 512 960 160 056 880 960 520 184 840 160 696 128 648 280 064 040 440 864 000 480 560 240 880 680 600 896 120 120 016 160 480 612 008 000 112 680 600 320 440 184 120 000 920 DIR'N
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4 24 20 34 38 15 31 11 99 102 121 100 ,1 151 101 35 128 316 1'8 159 291 180 325 648 156 235 530 464 548 180
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5 105 85 26 195 260 100 105 250 285 116 255 465 265
36 15 125 12 19 248 246 148 151 251 240 188 21' 481 235
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Table 3.
(continued)
AL6EBRR 86, REPRESENTATIONS Of CLASS 0 4
16
12 20
36 16
20
24 24 32 40
64 28 36 32 44 36
52 68 40 100
48
48 44 56 52
1 12 60 13
n
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12 12 52 56 80 104 60 56 144 68 16 64 60 16 64 84 80 68 68 100 108 12 84 S .P.
1
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1
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12 160 120 12 240 1'2 480 64 60 120
12 960 480 960 480 384 120 120 440 12 160 960 920 960 960 960
60 480 960 384 120 120 880 920
12 480 960 240 280 480 840 960 960 960 920 120 480 880 384 0 .s.
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28 29 30 31 32 33 34 35 36 31 38 39 40 41 42 43 44 45 46 41 48 49 50 51 NUnBER
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n
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24 25 26 21
54 69 100 85 105 15
66 204 21 124 230 250 429 m
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40 24 21 82
1 1 1 345 160 290 259 315 300 196 100 555 56 444 435 309 650 480 386 844 825 605 515 385 420 155 924 1
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94 21 54 110 195 189 220 215 65 130 224 240 1~ 81 620 325 21 414 390 no 415 455 82' 809 500 485 m
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6 183 1n 18 229 201 381 360 234 242 582 166 180 459 354
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38 39 30 56 44 141 83 96 61 159 40 45 209 106
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Table 3.
(continued) _. 00 00
AL6EBRA B6, REPRESEIITRTIOIS OF CLASS 1
6 14 22 30 30 38 54 38 46 46 54 54 62 86 54 62 62 10 TO 18 94 10 126 18 18 18 86 86 86 102 102 86 94 110 134 94 94 114 102 110 102 102 110 102 118 118 110 110 134 142 110 126 S.P. . 1 1 3 1 3 3 1 3 1 1 5 1 3 3 1 1 3 3 1 1 11 3 3 3 1 3 1 1 1 1 1 1 3 11 64 384 960 384 280 920 384 960 384 840 64 960 920 384 840 840 920 840 384 920 920 16Q 384 280 840 840 680 920 960 960 840 840 384 920 920 520 840 384 840 840 960 280 840 680 680 920 920 920 920 840 520 384 0 .s.
U 34 44 46 ~ 56 9
9
~
" " " 68 TOH~~ 16 16 18 80 80 UR U 84 B6 B6 86 H H H H
~
92 92 92 MM MM M 96 96 ff ff ff ff1001"UOUO U~L
1 1132112213294 163381316961099231 4 413~nn~rnnmmnmmmm~mfflmmmrumm~mrummmmmru~m~mm~mrurumm~~mm~ 64 168 160 992 312 840 296 456 608 128 456 600 200 360 552 200 384 632 864 160 412 056 552 6" 600 184 016 400 800 840 640 520 128 264 200 216 480 064 480 400 904 800 T04 488 800 600 656 152 216 316 200 120 DIn 'N
1 0 0 0 0
0 1 0 0 0
0
2 0
)
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1
0 0
1 1 0 0 0
3 0 0 0 0
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0 0 0 0 1
0 0, 0 0 0
)
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1
1
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2 1 0 0 0
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1
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5 0 0 0 0
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0 1 0 0 0
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4 0 1 0 0
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0
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0
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6
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0
UEl6HT
123456
20 21 44 100 56 s
"
16 259
1
~mmm~mruM~mmm~~~mrum~mm~mfflm*mQMmm~~™mmm~mmmm
14 35 21 25 30 104 21 80 135
1
S6mm~~mmm~~fflmmru840am™rummm~m~~mmm™*mmmmm~mm 1
12 38
30 so 21 82 136 110 184 R 180 no 315
S6m~ffl840mmmrn~~mrumrumammm~m~m~rn~mmm
5
14
15 35 21 25 60 30 104 21 115 135 126
S6~mrn•~mm~mm•m~0ru~m*~~*~840~W™™rum
2
12
4 10
6
4
4
2
2
9 15
0
1
1 1 1 1 2 261 114 158 284 321 210 300 205 005 529 56 141 681 360 433 863 506 605 996 146 195 810 859 468 118
4
1 1 9 45 50 51 21 46 82 80 216 110 61 100 184 145 82 215 1TQ 516 226 56 420 3'6 168 181 525 128 040 532 344 558 580 624 326 398
5
6 33 54 so 12 40 10 65 i44 21 99 181 219
5 10
6
24 12 38
5
6
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15 35
21
12 18 23 24 20 22 20 63
3
4
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4 14 15 24
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9 10 10 10
3
• 60 30
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21 115 135 126
15 30 54 56 65 50 20 40 15 61 39 25 225 150
56 100 195
• 165 164 480 240
• 120 285 424 285 111
6 160 150 101 140 181 424 330 280 246 385 159 110 680 251
n
20 33 38 88 50 31 40 12 11 40 85 65 243 121 21201181 91 101 248 316 484 261 198 302 210 284 103 210 10 24 16 30 30 12 48 12 24 12
6
35 104
1 42 18108160132 93 61 m 160 112 120 80 488 285 21 360 332 195 244 412 808 164 533 428 614 380 404 325 440
6 10
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m
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1 60 60 51 80 " 224 120 145 136 220 54 60 335 146
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60 50 114 50 21 112 82 44 40 156 160 296 110 92 122 200 184 365 82
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15 38
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n
80 80 216 110
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15 35
21
25
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10 30 36
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9
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12 18 30 23 16 13 24 33 20 21 20 100 63
61 135 184 145 82 21
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88 18 48 51 106 190 204 132 114 165 111 116 356 112
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Table 3.
( continued)
AL6£BRR 81, REPRESENTATIONS Of CLASS O -----------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------
U
4
12
20
36 16
24 20 32 40
24
64 28 28
36 44 32
2
3
52
68 36 100
40
40 48
48 56
2
3
2
1
69 69 11
11
11 13 13
ll
44 12 12 52 80 6
52
3
2
48 104 56 52 144 60 6
2
2
60 68
6
16
56
2
6
16
64 84 64 60 100 108 80 1 13
8
11 19 19 19 81 81 81 83 83 83 85 85 85 81 81 81 89 89 91 91
91
\.0
0
5 .P.
3
M84M~mM~~m84m~Mm~~~mmm~Mmmm~~m84~mm~mm~~wmm~~m~~~~~m~Mo~. 15
21
29
~1
41
43
45
51
51 53
55 5$
51
51 59 63 65
65
61
1
2
93
93
1
93
LEVEL
1
1 1 5 3 4 6 5 2 6 11 26 35 38 33 32 65 10 92 85 59 111 230 222 88 113 211 118 218 388 125 466 510 35 841 608 581 583 142 611 433 261 018 361 611 686 314 1 15 105 119 455 105 665 365 355 003 080 916 005 940 435 011 180 568 615 915 481 631 948 820 085 500 040 945 168 119 264 700 605 460 960 580 830 510 100 815 608 860 ~ 560 511 160 260 505 360 253 205 450 DIn 'N
2 0 0 0 0 0
0 2 0 0 0 0
0
0
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4 0 0 0 0
>
4
5
1
8
1 0 0
3 1 0 0 0
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0
0
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0
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9 10 11 12 13 14 15 16 11
0 1 0 0 1
>
20 21 22
23
24 25 26
21
28
0
2 1 1 0 0 0
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0
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29
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32
4
4 1 0 0 0 0
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36
31 38
39 40
41
42
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44 45 46
41
48
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UEl6HT
50 51
NUMBER
------------------------------------------------------------------------------------------------------------------------------------------------------------------------.,·--------------------------------------------1
13
1 21 63
13
18
15 12 19 15
20 66
92 114 15 105 91 135 28 140 140 80 162 215 364 93 186 405 m
408 424 m
11
12 19 10
10 21
21 34 60 29 25 65
m
21
48 10
35
28
35
13 4
11 18 15 4
1
1
3
1
1
11 98 120 115 105 91 18' 28 350 245 210 392 715 448 259 416 490 483 462 184 308 840 840 84 225 896 170 148 232 819 612 428 003 568 631 693 150
19 26
33
22
11 11
18 19 15
l 125
90
75 146 264 164 93 180 185 m
19 4
4
1 28 410 380 295 292 528 313 288 552 329 684 241 261 195
51 156 164 31 183 236
1 432 262 281 211 252 303 132 536 633 534 241 261 319 T 205 90 150 146 !14 112 54 330 214 195 165 180 185
50 24 49 82 138
15
10 10 21
4 21
39
33
4
14
43
21 55 151
11
4 11
120 112 355 345
45 11 34 60 35 64 50 25 65 10
42
H
13 4
1 1 1 1 28 105 620 125 682 632 183 328 317 563 204 631 612 925
45 20 35 66 143 54 51 114 15 126 6' 100 91 135 105 28 225 140 155 162 204 111 124 341 594 250 165 186 405
29 25 48 18 19
465 510
23 46 80 10 19 42
18
13
10 16
18 15 18 54 21
54 61 51
51 128
19 15
1 45 20 35 66
51
31 144 140
l 153 156
'7 118 113 552 290 13 81 334
95 94 216
11 11 33 22 35 22 38 25 48 40
l
10 50
48 49
82 55 50 103 235 103 49 55 151
98
1 145 110
92 90
92
30 10
25 21
30 32 24 59 90 35 34 34 60
53
60
61
22
12 11
24
10 1' 10 10
11
11
18
1' 10
10 16 14
10 15
14 50
51 51
91 48 170 228 232 13 80 135
14
15
8 15
6 19
4
4
4
14 20 6 21 4
43 45
11
35 64 21 20 82 21 44 35 208 118 14 15 125
12
16
11 21 42
10
23 34 60
15 10 10 21
13
4
18 114 15
• 143 54
48
10 55
4
35 49
14 35 64 98
21
20
28
21
46
33
54 19 101
11
4 21 21
66
33 41
4
4
28
25
19 12
31
18 15
93
8 58 34
83
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~
r+
13
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35
14
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54 61
15
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34 33
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4
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Table 3.
( continued)
AL6EBRA 87, REPRESENTATIONS Of CLASS 1 1 15 23
31 31 39 55 39 41 41 55 63 55 81 63 55 63 11 63 19 95 11 121 71 79 79 19 81 19 103 103 87 111 81 81 135 95 95 175 95 103 103 111
95 111 103 119 !03 103 135 143 119 S.P.
2 4 5 4 13 2 2 5 11 13 13 13 5 5 5 26 4 17 26 26 2 13 13 5 17 13 5 5 2 53 13 11 26 13 4 26 53 11 5 13 13 128 896 688 896 480 316 896 480 440 688 688 316 896 896 128 920 440 440 440 376 316 316 896 880 896 480 920 880 880 688 440 44Q 316 920 44Q 316 316 688 896 160 896 44Q 920 880 440 480 880 160 920 316 440 440 0 .s. 29 43 55 51 65 69 11 13 19 19 81 83 83 85 85 87 91 93 93 95 97 91 99 '9 99 101 101 105 105 101 101 101 109 109 109 111 111 111 113 113 113 115 115 115 111117 119 119 119 121 121 111 LEm 1 2 3 2 3 3 3 1 1 2 4 9 19 13 7 14 11 15 19 11 11 21 13 3 61 15 45 43 42 53 21 101 123 51 52 56 84 111u«ru11rumm•m~~mmm~~~mmmm~mmm~mmmm~mmm~mru~m~m1Mmm~rnmmm 128 192 520 440 800 688 680 480 200 664 032 440 256 6'IO 040 256 800 256 560 960 516 880 104 184 304 200 160 584 120 920 200 920 464 960 440 320 160 056 896 440 280 040 368 456 880 600 800 040 824 040 800 360 0 In 'N
0 0 0 0 0 1
4 0 0 0 0 0
1 0 0 0 0 1
5 0 0 0 0 0
0 0 2 0 0 0
0
1
1
1
1
1
1
> >
4
> >
4 1 0 0 0
>
4 0 1 0 0 0
1
1
>
)
9 10 11 12 13 14 15 16 11 18 19 20 21 22 23 24 25 26 11 28 29 30 31 32 33 34 35 36 31 38 39
5
0 0 0 2 0 0
'IQ
41 42 43 «
)
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45 46 41 48 49 50 51 NUn8ER
1 1 1 1 1 1 3 1 2 4 1 5. 2 5 6 5 6 6 3 1 4 19 4 13 12 12 15 7 26 33 15 12 13 10 n~10rn94m~m™™mmm~m~mmmm~~~rn~mm~~-~~mmmmm™mmmm~mmmm~•~ 1 1 3 1 1 2 3 2 3 3 1 3 2 9 2 6 6 6 7 3 13 11 8 6 1 11 ~48~«mnmm~84Hmm~rnmmmmrn~m~~rnmmm~~m•™m~~ffi~~~mmm~~m~~
u
11 11111 11 4133331684335 M~ZS4269~~n~m~~mmm84mm~ru~~rnmmm™ffimm~mmmm~~mmm~~mmm 20 4 15
21 48 11 18 12
12
28
4 4 1 2 2 3 2 1 1 1 2 « 100 110 69 183 210 16 84 259 51 195 392 910 444 392 18s 633 861 690 516 ,12 112 356 210 437 483 845 880 350 231 090 450 218 680 324 m 111 3 4 2 1 1 2 2 1 1 1 1 52 80 10S 101 99 90 140 28 210 103 116 332 660 554 247 415 660 486 134 722 322 848 532 84 111 620 530 440 611 120 993 145 392 168 516 625 808 14 35 55 2S 63 69 30 28 104 21 80 149 365 195 162 305 264 345 315 240 266 316 m 20
21 48
28
4 16 23 30 42 21 24 62 4 10 15 4
11 18 12
15
4 28 12 12
• « 100
• 210 76
84 259 51 195 392
. 315
• 910 444
• 518 185 633
l 102 49 65 124 240 242 88 110 288 111 318 341 111 401 268
~
1 2 1 1 934 313 654 616 717 130 480 330 112 116 634 680 341
30 52 130 82 51 105 101 120 136 110 90 1'11) 82
~
454 103 356 332 315 430 215 830 950 436 4« 475 660
38
33 24 20 40 72 102 25 50 120 48 I~ 165 30 194 138
6 12
14 35 13
10 16 38
21 S5 25 63 10 45 20
69 30 44 80
21
93 12 48
n:;-
1 310 161 252 240 361 218 220 500 976 572 232 250 620
15 14 70 25 42 55 25 84 60 30 69 30 21 28 208 21 195 148 126 265 100 500 390 164 324 305 264
14
12
56 110 69 183
4
2 2 1 1 1 1 1 84 081 222 840 816 660 025 ffi 005 049 860 010 110 458
~
104 21 80 148
• 135
• 365 1,s
• 218 305 264
1 125 85 81 80 162 87 95 155 432 288 68 15 210 ~
«
• 100
56 110 69
PJ
10 11
-a ,..,. (1)
-,
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12 13
14 5
12
4
4
42 42
• 35 49 21 20 63 21 35 35 168 148 14 15 105
14
16 38 33 16 30 42 33 54 50 24 62 40
15
1 114 49 135 124 144 160 99 305 423 219 160 110 288
15
4 20
11 15
4 10
6 15
14 35
9 23 24 12 18 12 11
16
18 12
9 15 52 51 95 46 180 164 8Q 114 105 101
16
18 105 101
11
~
51 24 42 40 63 49 42 92 180 108 46 50 120
18
!:!.
14
38
30 52
50
• 130 82
C
12
4
3
15
5
18 23
4 28 20
12
12
6
12
4
10
15 14
35
14 12 11
10 25
63 55 25
19
35
21 55 25
20
10 45
21
20
22
15 13 10 10 24 11 15 20 68 54
6
4
4
4
24
4 23 16 24 28 20 53 66 38 32 30 42 24 24
4
15 12 4
4
4
Z3
15
24
15 10 30 18 12 22 15 15
38 33
22 30 42
26
4
10
20
17 15
21
'
15
11 24 18
10
4
15 12
-I s,, CT
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28 29
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c=;·
~-
25
16
4
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4
6 15
4
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30 31 32
4
33
34 35 4
36 4
31 38
4
39 40
41 42 43 44
45 46 41
.....
48
\0 IJJ
49
50 51
Table 3.
(continued}
AL6EBRA 88, REPRESENTATIONS Of CLASS 0
4
16
f2 20 '36 16 24 32 20 40 64 24 28 28 36 32 44 52 32 68 100 40 36 48 48 40 S6 44 12 12 44 80 52 104 52 144 48 60 56 68 16 52 16 56 60 56 84 64 64 100 , .P.
16 112
1 1 1 1 4 1 1 1 8 6 10 4 1 2 2 1 17 8 4 26 13 10 1 4 21 1 17 1 1 2 1 2 16 448 224 16 120 344 112 792 224 16 792 480 024 344 256 344 224 960 224 16 720 152 448 480 168 688 048 112 344 920 224 960 224 480 16 880 448 152 344 480 504 344 920 168 168 688 120 048 224 0 .s.
11 31
33
43 41
49 53
59
61 61
63 65
61 69 11 73 13 15 11 11 19 81 83 83 85 85 87 89 89 91 91 91 93 93 95 95 91 91 99 99 101 101 101 103 103 103 f03 105 105 105 107 LE~El
12111212223131 1 2 9 6 6 11 4 12 33 19 50 24 61 63 90 59 19 209 184 131 259 293 493 369 186 367 596 312 716 255 124 69 259 152 484 465 448 040 646 306 382 586 049 311 001 456 11 136 152 680 615 952 380 044 183 188 340 692 316 592 448 388 310 413 308 448 432 380 304 156 784 896 930 681 512 200 080 904 232 040 000 812 168 100 472 406 128 655 114 008 220 380 584 800 840 368 084 O!n'N
4 0 0 0 0 0 0
1 0 0 1 0 0 0
0 0 2
0
0
>
15 15
28
0
0
0
0 0
>
14 15 16 11 18
19
20 21 22
84 63 28 92 36 S6 104 56 132 10 161 140 280 120 21
14 21 22
21
91
35 125
20
14
15
25
15 32 20
22
4 1 0 0 0 0 0
0
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9 10 11 12 13
0
1 1 1 0 0 0 0
15
24 25
26 21
23 10 31 10
4
4
30 31
32
33
34 35 36
1
40 34 81 29 21
21
21
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4
4
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39
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40 41
1
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42
43
1
1
44 45
1
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MEl6HT
46 41
1
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48
49
50
2
1
2
51
NUnBER
96 105 205 110 364 115 126 245 239 238 261 148 250
1 1 1 1 1 1 1 1 1 2 36 135 192 198 161 092 239 239 085 330 309 162 560 400 980 36 580 669 636 411 410 654 438 930 110 865 166 500 935 330
15
29 59 80
18
15 12 25
20
24
0
>
0 1 0 0 0 0 0
51
39 34 21 29 22
0 0 1 0 1 0 0
48 91 35 195 35 11 154 11 168 105 120 98 36 115 315 231 216 224 189 245 350 245 245 468 210 315 224
63
21
31
2 0 0 0 0 1 0
21
22
15
98 110 32 65 201
48 21 28 56 58
23
8 255 623 281 400 385 480 426 385 525 510 141 200 555 330 96 210 205 135 239 140 110 140 455
80 115 224
.m
• 35 105
81 29 92
21
21
21
22
32 64 68
60
16
36 11
50 10 20 39 21
41
21 29 21
60
85
15
58 59
65
66 125 80
90 123 15 110 58
26
28
22 11
15
36 15
30 25
15
38 45
20
10
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10 25
16
12 23
60 233 36 229 41
51
20 20 40 66
13
11
4
29
0 1 0 0 0 1 0
~rnmmru~~~mmmrnmffl™m~mrnmmmmmm•m~m•
15
4
28
>
35 154 140 105 120 36 195 231 111 224 245 311 315 126 252 609 238 612 148 611
13
13
23
>
6 0 0 0 0 0 0
60
14 20 39
13
10
62
20
10 21
15 291
63 21
67
48 91
10
• 35
8 209 188 232 116 114 2il0 120 339 263 340 208 182 372
20
40
80 180 88 113 110 183 118 120 202 191 205
62 216
87
51
20
21
22
18 15 12 25
19
15
15
4
20
4
4
16
21
15
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10
35
10 14
15
14
10
35
18 21 20 84 21 110 92 135 35 58 149 14
19
24
4 26
22
81
21
51
56
48
15
14
20
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21
21
20
30 23
14 4
11
34
50 121
10 11 12
13
15 10
81
24 60
58
n::::, p.J
13
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65
42
58
69
21
28
11 11 35 14 85 14
15
10
31 15 10 10 69 10 10 40
16
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n
n n
n
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Table 3.
(continued)
ALGEBRA 88, REPRESENTATIONS Of CLASS 1 -- --- -- - - - ----- -- -- -- - -- -- - -- -- - -- - - - -- --- ---- --- -- --- -- ----- ---- -- --- -- ---- --- -- ----- - ---- - --- ----- -- --- ---- ----- -- --- -- - -- -- ----- --- -- ---- ----- ------ ---- -- --- ----- ----- -- -- - - - -- - - - --- ----- - - -- -- --- - - - -- - - - - - - - - --
--Ii
8 16 24 32 J2 1iO 56 40 48 56 48 64 88 56 56 64 64 12 12 80 64 '6 128 12 1Z 80 80 80 88 88 104 104 80 112 88 136 88 116 88 '6 '6 104 112 '6 112 96 104 104 120 104 112 136 S.P.
I.O 0-.
2 1 2 14 14 2 11 43 1 14 14 2 1 11 2 43 43 14 11 14 2 101 43 14 11 14 86 2 1 43 143 14 11 14 11 2 101 215 43 43 11 43 43 143 14 1 86 11 2 14 256 048 168 048 336 336 048 920 008 168 336 336 048 168 680 048 008 256 008 336 680 336 048 520 008 336 680 336 016 048 168 008 360 336 680 336 680 048 520 040 008 008 680 008 008 360 336 168 016 920 048 336 0 .s. 31 53 61 69 19 83 85 89 95 91 91 99 101 103 105 101 109 109 111 113 113 115 111 119 119 121 121 123 125 125 121 121 121 129 129 131 131 133 133 135 135 131 131 131 139 139 139 139 141 141 141 143 LEVEL 1 1 2 1 5 1 9 1 12 12 n 11 3 34 22 22 43 26 ea 18 34 " 84 " 103 48 111 13 145 339 114 240 233 m m 305 203 120 518 185 141 2s1 4~34rn™mffi~mrnmm~~~m~mmmm~~~rnmm~mrn~rn~mmrnm~~~mmmru~romillmmm 256 095 464 816 264 064 896 200 560 920 848 264 256 129 088 432 056 rn 488 600 240 116 024 208 160 824 200 112 384 320 680 400 512 352 624 544 612 584 880 ooo 928 456 ooo 040 408 848 232 128 136 812 608 m 01n•w
----------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------0 0 0 0 0 0 0
0 1 0 0 0 0 0
0 0 1 0 0 0 0
1 1 0 0 0 0 0
0 0 0 1 0 0 0
1 0 1 0 0 0 0
0 2 0 0 0 0 0
0 0 0 0 1 0 0
2 1 0 0 0 0 0
4 0 0 0 0 0 0
0 0 0 0 0 1 0
0 0 0 0 0 0 1
0 1 1 0 0 0 0
0 0 0 0 0 0 0
2 0 1 0 0 0 0
1 2 0 0 0 0 0
0 1 0 1 0 0 0
3 1 0 0 0 0 0
1 0 0 0 0 1 0
0 0 2 0 0 0 0
2 0 0 1 0 0 0
1 0 0 0 0 0 1
0 3 0 0 0 0 0
0 1 0 0 1 0 0
2 0 0 0 1 0 0
4
6 0 0 0 0 0 0
1 0 0 0 0 0
1 0 2 0 0 0 0
1 1 0 1 0 0 0
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2 0 0 0 0 0 1
0
2 1 0 0 0 0
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1
1
1
1
1
1
1
1
1
1
1
1
1
3
1
1
1
1
1
1
1
1
1
1
1
1
1
1
1
4
5
7
8
9 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24 25 26 27 28 29 30 31 32 33 34 35 36 37 38 39 40 41 42 43
35
1
2
3
3
3
t
1
1
1
2
6 11
UE!GKT
44 45 46 47 48
49 50 51 NUftBER
1 21 48 23 32 30 22 34 38 24 14 61 21 16 28
l
4
4
8
4 16
3
5 11 14 10 16
3
1
1
3
2
1
2
4
6
4
l
3
1
9 22 11 14 14 10 16 18 11
1
2
2
2
3
1
3
4
'
5
6
6
4
1
8
5
3 13
1
1
1
1
1
4
1
3
3
1
4
3
2
1
1
1
4
2
2
2
2
3
3
2
1
1
1
1
1
• 410 18' 343 700
• 546
~mmrnmm~mmmru~mm~~mrummmmmmm~m™mrn~mmmm~mmm~~mmm~mru~m 27 63 36
1
14
1
l
29 10
l
13
4
3
6
8
2
1
4
5
2
1
2
1 ~rnmmmm~mmmmmmrummmmru~~rnmmm~m~mmm~rnmmmm~ru~~ffl~mm 20 75 56 44 91 36 69 240 76 358 27 28
63 36
70
• 154
14
20
1
48
1
63 36
21 28 20 12 28
36 32
12
"
28
18 20 14 21 14
4
1
14
20
4 6
4
1
• 10
• 154
2
1
1 • 568
• 121 868
4
~~~mrummmmrnmrum~mrumm~~~rnm • 84 258
• 213 133 308
• 120
.m
1 144 149 1,1 m 119 230 36 264 155 333 573 545 316 682 663 392 222 306 430 213 619
55 25 42 15 30 180 21
n
20
56 15 44 112 44 91 100 36 -69 340 69 310 240 16 414 259 16 51 118 195 51 490
21 20
96
42 51 25 50 14 88 58 30 60 164 51190 35 195
14
20
48
28 15 21
4
3
2 1 1 1 1 1 1 1 8 150 132 93 180 110 336 124 120 234 448 231 524 141 552 36 862 580 993 012 020 084 116 6'0 240 '184 116 ,,0 893 980
18
3
1
~~mm~m~mmm™mfflfflmmmmmmrummm~~~™~m
75 84 44 91 36 110 69 121 240 16 568
8
4
2
.rnmmmmfflmffim~~mmru~~mmfflmrnmmm~~mfflm~~mmm
25 15 30 111 21 144 133 192 119
18 13 14 21
1
~~fflmmm~rum~m~rnmmm~rum~m~rumrnmmrn~m~m~mm
15
10 33 16 28 11 20 12 46
10 52
84 44 91
28
19 36 52 39 66 28 80
• 170 69 121 240
• 210
16
• 568
51 329
11
84
12
1 150 160 111 102 100 234 108 212 268 192 1,0 153 219 82
13
36 58
10
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36
63 60
8 365 550 420 365 350 505 3'6 101 517 383 105 406 431 319
n:::, I»
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8 101 255 132 192 180 140 225 210
• 154
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101 426 116 124 210
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14 13 18 14 27
24
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55 35 63 11 10 111 21 91 126 98 35 49 113 14
15
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35
25 110 25 105 15 30 132 104 30 21 249 80 11 154
16
20 18
12
4
4
28 56 56
48
21
• 180
21 105
18
96
84
19
62 60 102 74 49 112 65 58 55
20
20 14 21
55 25 42 75
69
14
20
21 20
48
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16
4 23
42 72 51 52 50
14
17
14 64
30
20
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28
21
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38
13
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10
30
28 20
12 34 38 12
15 15 18 10 10 28 11 33 33 24 20
4
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18
18 14
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12
12
20
28
36 15
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25
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26
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2 0 1 0 0 0 0 0
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z
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----------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------11 11
36 108 80 36 m
2 2 1 2 I 1 4 3 3 2 1 3 1 1 1 1 1 1 1 45 84 135 111 84 208 180 126 420 153 126 4 5 ~ m ~ m r n ~ r u ~ ~ ~ m m r n r n ™ m m ~ ~ ~ m m r u m m m m m
24 16 28 25
1 2 1 1 I 2 1 1 I I 1 2 3 28 120 163 56 200 180 56 140 153 70 45 259 141 364 296 495 164 392 328 944 308 616 036 189 021 630 45 168 116 143 260 6@ 844 112 366 1'9 880 428 520
23 16
21 29 31 21 46 3' 35 119 33 35
25 11
24 24
15
21 36 15 45
28 25
39 15 32 33 20
23 24
15
25
1 1 1 1 1 9 238 1@ 133 214 480 164 281 320 328 308 301 364 189 341 385 45 952 903 560 036 548 584 496 106 015 1'6 433 1'0 84 63 28 120 255 100 56 200 104 216 56 140 153 132 10 45 280 420 281 364 296 324 552 308 615 328 m 60 34 108 6' 114 31 m
21 14 21 29 59 23 21 46 91
11 4
10 29
4 10
15
4
24
, 105 392 280 133 214 311 1'5 231 600 111 84 63
28 25
28 120 92
• 255
2n
• 136 56
21 58 23 15 45 25 41 15 32 33 32 20
Bl 100 68 108 6' 11 111 96 143 105 61 117
29 10 16 15
23
32
32 10 16 23
25 13 68 85
24
81 10
21 42 35 21 29 44 63 21 81
4
21 10 14
29 15
14 45
23
10 29
4 31 10
4
4
14
13 23 15
29 22
59
31 21
14 24
10 11 12
15
17 10 16
48 46 21
I 100 215 134 10, 130 139 148 295 240 16 101 322 21 14 21
25
4
281
9 310 223 131 341 134 141 490 367 243 260 101 3'8
30 16 4
24
41 35 119 33 125 35
m
9 366 849 540 400 51' 510 546 852 993 282 433 923
51 33 33 66 112 31 90 14 82 6' 95 91 41 85 130 20 14
4
15 290 6' 205 324 41 316 210
20 21 138 111 35 12 14 120
13
29 67 90
14
40 35
21 43 18 15 80 12 46 15
15
20 35 21
21 20 21 28 96 35 14 14 105
16
90 35 21
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21 96 61 33 66 16 51 15 141
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Table 3.
(continued)
ALGEBRA 89, REPRESENTATIONS OF CLASS 1
9 11 25
33 33 41 51 41
N 0 0
49 51 49 65 89 51 51 65 65 13 81 73 65 91 81 129 13 81 13 81 89 105 81 105 81 113 89 89 131 89 97 111 89 91 105 91 113 113 97 91 121 105 137 105 S .P.
4 10 4 43 36 4 64 m 10 64 36 4 43 250 m 10 m 36 4 322 36 4 381 43 258 258 258 1s 129 129 645 36 36 m 36 258 4 4 645 114 129 258 258 129 645 387 258 64 36 m 512 608 432 608 008 864 608 512 024 432 512 864 608 008 048 024 432 024 864 608 560 864 512 608 072 008 048 048 048 432 024 024 120 864 864 560 864 048 608 608 120 144 024 048 048 024 120 072 048 512 864 024 0 .s. 46 64 80 82 94 98 100 106 112 114 116 116 118 124 124 128 130 130 132 134 134 134 136 136 140 142 142 142 146 148 148 148 150 150 152 152 152 154 154 154 158 158 160 160 160 162 164 164 164 166 166 166 LE~EL 11111112111 1 5 4 3 7 3 7 22 34 11 46 43 12 62 41 8 13 148 95 127 185 363 136 193 267 437 273 211 525 187 549 141 51 928 637 132 093 102 278 805 446 368 059 143 685 911fi~m•~rnm~~~mmmm~mmmmm~mmmmmm~m~mrnm~mm~mmmmmmmmmrnm~z 512 216 824 552 576 040 680 384 160 816 024 368 320 048 312 384 072 568 264 328 000 012 552 008 952 400 768 120 584 328 920 992 896 656 280 000 960 144 608 864 616 400 672 240 880 728 920 248 496 856 912 192 OIM'N
t
0 1 0 0 0 0 0
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4 0 0 0 0 0 0 0
0 0 0 0 0 0 0
t
1
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> >
t
0 1 0 0 1 0 0 0
1
I
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4 1
>
9 10 11 12 13 14 15 16 11 18 19 20 21 22 23
0 0 0 0 0
0 0 1 1 0 0 0 0
0 1 0 0
I
t
I
0
0
1 0 0
> >
1 1 0 1 0 0 0 0
> > I
24 25 26 27 28 29 30 31 32 3' 34 35 36 37 38 39 40 41 42
2 0 0 0
3 0 0 1
Q
0
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0 0
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2 1 1
0
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43 44 45 46 47
48
UEI6HT
49 50 51
NUMBER
I I I 3 4 I 5 5 I 1 4 I 14 8 II 17 30 10 14 20 33 20 14 38 12 38 9 3 58 104 68 64 64 72 99 71 128 54 57 85 4445mmm~m~mmffi~~mm~mm~~mmmm~mmmmmmmmm~•~~™m~~~mmm~m™m 1 1 2 t 2 I 5 3 4 6 12 4 6 8 13 8 6 15 5 16 3 I 25 44 29 28 28 31 44 34 56 24 25 38 n~•~m~mmmmmmmmmmrnrnmffimmmmnmmffi~mmm~mmmmm~448~m~~mw~m u
2 1 1 2 4 t 2 1 5 3 2 6 2 6 1 10 10 12 u 11 13 19 e a u u 11 21~12~m45rnmmmrn~mmmmmmrn™~mrnmm~rnmmmm~mmmmrnmmrn~mm~rn 35 36 21
80
15 20 24
45
• 104 224
• 371 388
• 230 432
2 1 2 3 9 6 4 6 7 7 4 15 4 7 6 2 1 2 1 2 I • 165 783 560 392 152 520 038 518 107 904 232 512 630 680 688 323 495 452 111 552 592 792 714 512 752 064 585 488 114
2 4 7 4 5 4 5 8 6 9 4 4 1 1 I 2 I I 2 1 44 102 141 69 189 172 76 310 152 51 4 5 m ~ m r n ~ ~ m m m m m m m ~ m m ~ m ~ m ~ ~ m m m w ™ m
16
27 63
98 108
10 116
35 36
80
1 3 2 1 2 2 3 1 5 1 2 2 I 45mmrnmmmmrn™~mmmmmmmm~™~~mmrnmmru
14 24 33 25 42 37 30 95 32 21 14
4
4
21
23
20 24
16
16 12 25
• 104 224 120
45
. 371
• 388
• 230 432
• 165
21 28
27 126 72
98 TO 144
27
4
1 • 368 518
63
36
24 10 16
10
. 833
. 472 189
11
. 224
. 120
12
. 108
70 116
45
35
36
80
45
60 10 66
21 20 31
n:::,
1 1 1 9 670 765 499 760 480 535 198 239 945 673 420 395
98
18 13 14 24 54 22 25 42 75 45 28 15
1 I I I 1 45 525 161 862 658 830 015 225 822 932 170 984 011 1 45 133 516 462 133 362 609 470 189 141 343 704 189
70 116 154
51 30 80 60 104 35 157 TO 240 66 200 275 40 276 145
16
2 . 520
I 2 1 1 I I 3 2 3 I 1 3 9 207 126 236 246 448 157 402 308 740 300 468 855 188 876 324 45 758 461 635 999 564 771 200 919 157 815 432 291 75 56 44 102 238 100 69 189 240 208 16 310 152 358 51
4
1 • 783 560 392 152 882
• 258 189 133 362 308 • 104
96 577
13
30 95 32 111 21
9 192 351 259 236 246 301 450 333 567 284 285 401
14
20 24 48
9 44 150 140 44 102 115 170 69 329 127 200 69
IS
86 75
72
67
1 240 195 U3 210 120 129 410 513 225 225
p.,
uM
~ ....,
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21 14
14 21
38 15
14 32
23
20 24
16
16 17
16 4
20
4
12 25
75 35 21 84 20 21 119 207
35 63 14 231
75 56 44 102 91
. 238
• 136 69
27 28
. 126
• 108
20
27 63
28 75
21
66 87 62 80 60 71 150 132 132 93 64 157
4 28
27
16
27
4
5
4
4
42 21 30
21 20
12 21 4
28
14 36 33 14 24 40 55 25 75 42 44 25
12
20 15 10 24 10 11 42 51 20
25
19 20 21
36
63
11
18
84
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16
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24
26
30 25
27
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28
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25
54
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0
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1
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66mm~mmmmmmrumrummru~
10 10 21 11
10
1, 30 30 31 11
35
46 51 4, 41 11 28 42 18 Z8 44 8' 1'2 146 56 41 11
45 135
21
" 1,0 45 n,
4
10 39 15 21 26
4
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• 155 310 330
18 31 36 15
10 10 21 11 4
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UEJ6HT
4, 50 51 NUnBER
2
2 10
2 2 5 1 3 3 2 3 2 6 1 2 2 3 115 014 415 800 42, 230 480 585 655 10 8,o u5 860 600
1 41 11 36 1'8 381 84 225 442 225 160 84 255 500 415 126 286 126 548 66 660 603 565 416 108
31 36 41 58
ll
0
0 0 0 0 0
0 0 0 0 0
66mru~~mmrn~~~mm~™~mmmm™mru~~~m~mrnmmm
2' 20 31 11
1' 29 30
4 0 I 0
0 1 0 0 0 1 0 0 0 0
23 24 25 26 21 28 2, 30 31 32 33 34 35 36 31 38 3' 40 41 42 43 44 45 46 47 48
1
10 21 11 10 30 20 31 11 45 1,0 45 254 310 215 231
> >
0 2 1 0
2 1 1 0
3 0 0 1 0 0 0 0 0 0
u
19
• 231
'°'
3 2 2 1 I 884 112 ,3, 211 618 302 681 745 232
1 • 110 66 120 225 215 615 120 45, 411) 600 515 165 '" 210 455 510 160
8l
1 1 56 51 10 4'8 11 212 ss1 532 406 532 883 841 648 ,21 101 5,0 560 681 135 ,52
29 58 59
41
• 153 11 36 1'8 206 639 84 450 442 180 506 225 950 84 455 500 332
10 30 30
31
,s
11
• 45
• 135
•
"
1,0
• 145 45 3,,
4
• 210 310
86 45 144 21 41 94 81 35 51 35 110 11 115 121 113 293 189 11, 114 512 183 622 313 39, 12, 141 816
18 35
2, 51 59
41
38
20 29 40
31
11
1l 54
19
10 29 30
ll
11
21
10 10
21
11
10 41 16 15
10
15
10
10
20 104 4
4 10
33 11 28 42 42 130 28 88 8' 156 '9 1,2 183 56 85 ,5 308
14
16 4
4
46 m
45 31 36 56 36 103
n
58 58 108
18 31
28 36 15
39 58
10
10
1,
10 30
11
109 285 1so m 111l 140 181 151 310 m 129 140 204
12
31
39 41 126 21 81 86 31 98 45 181 21 85 94 6'
13
10 14
41 15 14 35 45 21 20 130 21 141 35 120 14 15 1'4
14
58 51 24
15
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45 35
55
8 102
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Table 3.
(continued}
AL6EBRA 811, REPRESENT9TIONS OF CLASS 1
------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------11 1' 21
35 35 43 59 43 51 59 6l 91 51 59 59 6l 15 83 99 131 61 1 2 1 1 22 331 225 22 615 013 112 225 22 946 103 946 013 013 225 225
n 4
15 l5 83 83 15 91
83 101 101 115 91 139
4
2 5 2 22 615 130 055 331 331 103 n1 021
1 013 225 u m
"
83 119 83
91 91 99 91101 115
9
4
4
8
5
9
91115 '9 123
99 139 141
99 S.P.
N 0 00
1 9 2 8 2 1 2 14 1 5 2 1 2~.ummmmmmmmrnrn~mmrn~
2 22 m m 112 ~ru~~mmru~m~mrurn~rnmmm~ru~m~~mmm~~~mm~m~mruammmmm~~mm~mmmm o.s. 61 Hmmmrnrn~~rnmmm~mrnmrnmmrnrnmmmmmmrnmmfflmmmmmmmmmmmwmmrurummmm LE~EL 2
1
2
2
4
1
3
3
2
9 28 10 1' 18
24 21
Zl
38
13
11 1' 43
~mrnmmmmrnm~~w~mru~mmmru~mmmruffl~~mmm~mrnmm~m~m o,s 111 342 184 841 on 464 824 058 230 442 368 332 451 442-648 008 388 228 m 736 364 811 009 235 296 232 110 826 m 201 895 802 m ~m~™~mrnm~mmmmm~m~~~~mmm~mmffl™~mmru~mrufflfflmmmm~m~mrum~m~mm z
3 1 4 14 53 39 64 25 45 411 518 108 208 145 ~8 415 424 768 901 TI8 289 826 608 935 511
DIM'N
--------------------------------------------------------------------··-------------------------------------------------------------------------------------------------------------------------------------------------0
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0
0
0
0 0 0 0 0
0 0 0 0 0
0
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4
3 0 0 1 0 0 0 0 0 0
0 0 0 0 0
0
I
2 I 0 0 0 0 0 0 0 0
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5 0 0 0
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I
0 1 0 0 0 1 0 0 0 0
2 0 0 0 0 1 0 0 0 0
1 3 0 0 0 0 0 0 0 0
4 0 1 0 0 0 0 0 0 0
> I
UEI6HT
234561
2 1 2 1 2 8 4 11 14 12 10 3 6 23 46 9 21 53 49 92 10 31 61 58 , 35 5 132 8 85 141 144 385 122 245 234 296 251 324 446 142 193 208 411 1165"~mru~rurnmm~~mmmm~mrum~~mmrummmmmm~mmm~m~ruw~mrnrum~mmffl~ 2
10
11 54 120 9 20
1
3
4
4
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1
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1
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15
3
9181132
3112120
3 tz
2 48
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3 31
54 53 146 41 93 90 114 '9 121 114 51 16 82 186
"m~~mru~~~rnmmmmmrnm~m~m~mmmmmmmmm~mmm~m~m~~~~m~m II 44 153 110 115
10 11
54 55 120
2
5
5 10
3
1
6
1 11 1' 18 51 18 33 32
43 35 41 63
22
21 30 ll
66~ru~mmm~rum~m~mrum~~mmmmmmmm~mmrnmw~mmmmmrumm 66
• 2IIO
• 420 684 105 110 286
2l 19 30 11 35 168 104 234 2'1 265 230 10 20 11
44
1 • 560 826
I
1
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2 4 4 • 255 5,, 185
3 • 520
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66ffl~~m~rum~mrummrummmru~mmmrnmrummrumllirn™~
'9 1.53 165 115 66
• 141 512
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I 2 I 5 • 215 640 242 610
1 • 366 112 36' 581 • 200
• 420
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• 990
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1
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• 560
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1 1 5 • '80 242 610
. ,10
1 I 1 1 5 2 3 3 5 3 5 6 3 2 2 8 1 32 21 43 54 41 1111 11 lO 168 348 133 201 408 510 120 189 246 486 465 196 285 121 638 66 390 862 850 068 548 280 152 213 492 892 144 549. 110 934 855 8
18 21 29 30 11
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90 168 104 318
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55 153 165 54 55
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2 I 1 1 3 1 2 3 I I • 518 66 230 665 136 348 392 696 656 521 968 890 114 518 840 019 216 • 141
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1 I I • 300 269 225
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2 1 1 1 1 I 51 434 11 450 490 480 300 924 830 8IIO 615 816 866 530 389 664 120 115
12
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1 135 1os 100 265 324 165 160 480 111 531 2,s 540 124 13s no
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3 12 12 21 16 12 16 30 10 14
12 16 13 26 18 13 18 1' 24
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3
4
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Table 3.
(continued) N
w
ALGEBRA C6, REPRESENTATIONS Of CLASS 0
4
12 12 16 20 32
0
16 20 24 24 24 28 28 36 36 40
28 52 32 12 32 40 36 36 40
40
48 44 44 40
48 56 44 52 44 48 64 60 60 68 16 48 52 80 52 52 100 56 56 S .P.
1 1 3 2 1 23 31 51 15 60 12 240 64 480 60 120 12 960 440 160 960 960 960 384 60 480 120 920 120 240 12 840 960 960 880 192 440 64 480 880 840 480 960 960 960 840 920 60 960 384 120 480 160 920 120 440 160 120 960 960 0 .s. 1 21 23 33
3l 39 41
43
45
41 53 ss 55 Sl 59 59 61 61
63 63 65 65 61
61
69 69 11 71 13 13 15 15 15 Tl Tl Tl 19 19 1' 81 81 81 83 83 83 83 85 85
85
Bl 81 81 LE~El
2 1 3 1 2 2 121 6Uu~~S1nM45~~~119urnrn92m~m40mrn~m~~m~m™m~mfflmmm~mmmm 1 65 18 429 429 002 650 925 365 006 954 650 025 450 344 021 310 160 051 010 615 686 316 992 150 950 800 018 9'8 898 430 615 930 616 650 ~ 550 464 115 241 464 408 510 459 450 858 400 666 094 930 320 160 DIM'N
1
0 4 0 0 0
4 0 0 0 0
5 0 1 0 0
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(
(
0
0
1
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43
44 45 46
4 0 0 0 0
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4 1 0 0 0
1 0 0 1
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1 0 1 0 0
2 0 2 0 0
4 0 0 1 0
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1 0 0 1
0
0
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0
1
0
0
1
) CT
ro "'
28 2'
4
3
4 5
2
2
30
'
31 32 33
3
34 35
36 3
4
31 38
39 40
41 42 43
44 45
46 41 48
49 50
51
N
.w
.....
Table 3.
(continued) N
w
AL6EBRA C6, REPRESENTATIONS Of CLASS 1
2
6 10 18
10 14 18 18
22
N
26 34 50
22
26
26
30 30
34 38
38 34 34 42
42
50 54 42
58
38
74 42 42 98
'16 46 58 46 54 50 50 54 50 58 58 62 54 66 66 78 54 58 10 S .P.
1 21 1 5 1 32 3 2 23 23 S1 12 160 120 12 192 960 480 384 480 120 120 12 920 960 960 880 280 480 960 480 960 920 %0 384 120 480 384 120 160 120 960 920 11 840 880 960 840 160 880 960 880 840 384 960 880 840 480 480 960 160 920 960 12
28 32
34 36
44 48
48
SO
52 54 56 56
58
60 64 64 66
66
68 68 68 10
70
12 12 12 14 74 16 16 16 18
78
80
80 80 82
82
82
84 84 84 86
86 86
88
88
8B
88
88
11 I 21 222121313 4 1 4 11 11 11 4 27 35 43 144 57 112 1115 136 104 168 265 116 144 155 89 180 608 112 448 551 31 021 237 523 864 448 867 743 274 750 401 306 501 574 521 068 089 615 839 390 12 208 560 364 572 368 800 516 088 440 648 368 456 100 680 144 200 320 600 136 104 168 200 480 144 584 232 180 400 320 448 616 824 020 600 600 864 800 008 600 480 320 544 304 928 208 520 560 088 672 200 816
1 0 0 0 0
3 1 0 0 0
0
0
(1)
..,......
3 2
'4
4
3
3
8
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Table 3.
(continued}
ALGEBRA C9, REPRESENTATIONS Of CLASS O
N
-----------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------
.j:,.
4
N
8 12 16 20 32 12 16 16 20 24 24 20 28 36 36 24 40 52 72 24 28 28 32 28 32 32 40 36 40 32 44 44 48 56 36 36 52 36 64 44 60 40 40 68 16 48 80 40 100 S .P.
2 2 5 2 20 12 8 32 4 2 4 80 32 40 2 64 80 18 20 24 12 161 2 24 2 8 43 193 4 80 32 4 4 36 2 4 322 I 144 18 016 016 144 288 18 316 304 160 096 672 064 256 032 144 016 608 288 288 18 640 256 320 016 512 640 432 160 192 096 280 016 192 016 064 008 536 032 640 144 256 032 032 864 288 016 608 288 560 288 O.S. 1nnnu~n~n~~"""~~""""rnmmmmwmrnrnmmmmmmmmmmmmmmmrnrnrnmmmmm~ 1 2 3 2 3 9 3 6 13 11 12 10 21 15 10 9 32 21 22 5 20 23 12 24 12 9 15 10 102 S 211 BM 5 e z s m ~ m m m r u m m m ~ m r n m r n m ~ m ~ m m m m m ~ m ~ m m ™ m m m ~ w m r u m m ~ m m ~ I 152 111 901 305 568 364 985 504 194 125 811 200 160 910 632 444 291 134 888 080 941 968 645 160 613 146 120 936 750 312 320 800 860 925 944 594 048 448 336 352 300 349 128 823 400 400 561 310 000 584 350 OIn 'N
4 0 0 0 0 0 0 0
0 0 0 0 0 1 0 0
< < 0
0
0 1 0 1 0 0 0 0
1 0 0 0 0 0 0 0
I
1 1 0 0 0 0 0
25 26
10
21 3
3
5
11
4
4
4
4
4
28 29 30
12 4
4
4
4
31 32 33 34
4
4
35
3
36 31
38 39
40 41 42
43 44
45
46 41 48 4/)
50 51
I',)
Vl
Table 3. ALGEBRA
en,
(continued) REPRESENTATIONS OF CLASS 1
N
2
V,
6 10 18 10 14 18 22 26 34 14 50 18 18 22 22 26 26 22 30 34 38 38 42 26 50 54 58 26 14 30 98 30 34 30 34 34 42 34 38 38 42 42 46 34 46 58 54 SO 38 54 58 S.P.
1 14 21 3 3 42 28 m 2 141 n 21 m n 3 21 22 320 440 22 184 120 960 960 440 440 240 22 160 408 048 840 ,20 120 '20 360 960 120
''°
3
N
2 3 881 295 5,1 13 on 881 2s3 m 112 443 22 141 21 2,s 365 63 n 1 63 365 63 1 '20 640 440 '60 440 040 440 680 22 360 ,20 160 040 440 408 640 520 528 840 120 680 440 360 ,zo 320 360 440 360 '20 0 .s. 1 112
22 58 62 64 86 ,4 ,8 100 102 104 106 106 118 118 122 126 129 130 134 134 136 136 138 140 142 142 142 t44 146 t46 t48 148 t54 158 t58 158 160 160 162 164 164 166 m 168 no 110 no 112 m 114 114 m
LE~EL
1 4 4 4 4 13 , 9 10 18 6 8 9 9 26 5 30 1 64 68 62 163 10 '9 40 321 45 262 181 451 26' 431 1'9 128 613 654 616 581 132Nmm~~m~~~mmmm~mmmmmmrumfflm~mmm~mmmm~mfflmm~mmmmm~•mm 22 518 520 024 194 536 750 480 230 496 210 780 876 552 012 010 000 600 368 984 650 000 472 250 384 488 124 600 500 400 000 040 000 440 630 438 000 600 500 150 640 '36 680 650 000 000 500 250 336 152 600 328 0In'N
0 1 1
0 0
0 0 0 0 0 0 1 0 0 0
0 0 0 0 0 0 0 0 1 0
0 1 0 0 1 0 0 0 0
0
0
0
0
1 1 0 0 0 0 0 0 0 0
0
0
0 0 0 0 0 0
0 0 0
0
0
J
"O
....
(1)
4
5 u u
4
2
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n u u n u u u u 4
5
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n
3
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~
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Table 3.
(continued)
AL6E8RA 01, REPRESENTATIONS OF CLASS 1 4 12 20 36 20 28 28 28 36 44 36 52 68 100 44 52 52 52 52 60 60 60 '8 16 60 16 84 '8 68 100 108 16 84 16 116 148 16 84 1'6 84 84 84 100 84 100 92 92 92 92 116 100 100 S.P. 2 2 63112 6 28 1313 222 202 266 6 13116322 14 280 168 14 612 64 64 240 840 840 688 168 168 14 120 360 344 344 240 448 448 120 840 240 '60 840 680 360 440 168 840 240 688 240 168 168 160 688 14 240 120 120 448 120 448 440 344 344 120 360 688 688 0 .s. 13 31 35 31 41 43 43 49 53 55 55 51 59 61 63 65 65 65 61 61 61 11 13 13 13 15 11 11 11 19 19 19 19 19 81 83 83 83 85 85 85 85 85 85 85 81 81 81 89 89 89 89 LEPEL 1 31 233 3 5335255 2 1111U24362423 8 m m m m ~ ™ ™ ~ m ~ 6 4 8 w m m m ~ ~ m ~ m ~ m r u ~ 6 8 ~ r u m m m m r u m m ~ m m m 14 364 896 546 002 116 11' 648 200 948 608 024 296 008 128 950 830 830 012 140 140 504 808 000 640 408 500 128 164 384 856 180 800 180 462 568 920 664 952 352 696 000 858 000 858 950 094 094 480 800 508 508 DIn 'N
101 0 0102113 000221130020 14030351071010 02 4 0 0 0 1 0 0 2 1 1 1 0 0 0 1 0 0 0 2 1 O 3 0 O O O 2 1 0 0 0 0 2 0 0 O 1 O 0 0 1 0 1 0 0 0 0 1 1 0 0 0 0 0 1 0 0 0 2 0 1 1 1 0 0 1 0 0 0 0 0 1 0 0 0 0 1 0 0 0 O 0 0 0 0 0 0 0 1 0 0 0 0 0 0 0 0 0 0 1 0 0 1 0 1 0 0 0 1 0 0 0 1 0 1 0 0 0 0 0 2 0 0 0 0 0 0 0 0 1 0 0 0 000000000100200000001000000001010020000020000000100000011000000010200000011100000100102030010020000t0020 G O O O O 2 0 0 0 0 1 0 0 0 0 0 2 0 0 2 0 0 0 0 1 0 0 0 1 0 0 2 1 0 0 0 0 1 0 0 0 2 3 0 1 0 2 0 0 0 2 O UE~HT
1
2
4
5
6
T
8
9 10 11 12 13 14 15 1' 11 18 19 20 21 22 23 24 25 26 21 28 29 30 31 32 33 34 35 36 31 38 39 40 41 42 43 44 45 46 41 48 49 50 51 NUnBER
---------------------------------------------------------------------------------------------------------- .-----------------------------------------------------------------------------------------------------------12
15 10 10 60 80 96 120 81 12 4
18 24 30 44 26 22 10 16 10 16 11
1
' 2
1'
5
4
' 3
3 2
1
3
1
4 2 2 4 1 3 3 3 2 2 3 56 094 916 090 601 090 601 865 310 310 680 995 834 834
1 1 1 1 1 1 2 1 1 2 1 2 2 6 120 135 96 9 6 W ~ ~ ~ m m ~ ~ r u w ~ m r u m • m m r n m m 21 031 4'8 591 324 591 324 448 236 236 332 026 0411 040 6 40 55 30 30 54 40 40 114 124 125 155 140 210 110 435 131 146 160 290 160 151 81 880 315
5 12
4
2
~~mmmm~~~~ffl~m~~rnru•m~~rurumm
15
10 10 39 30 60
40 120
85 60
"
• 120
• 102 12 165
n n
1 1 1 1 524 116 140 146 140 146 320 595 595 210 uo 050 050 '4 116 130
• 130
• 480 135 135 320 330 360 360
44 50 41 41 48 46 46 130 82 80 188 86 150 18' 465 64 10 196 216 1'6 68 32 858 316
1 1 1 1 6 484 6,0 815 116 815 116 159 636 636 092 490 061 061
14 15 21 15 14 21 21 35 21 20 84 21 35 84 210 14 15 105 126 91 14
6 318 u,
1 210 300 420 105 405 ,1 504 336 315 462 210 546 546
14 15 15 21 14 21 21 35 21 20 84 21 35 84 210 14 15 91 126 105 14
6 318 196
1 210 300 405 91 420 105 504 315 336 462 210 546 546
14 18 12 12 u 4
42
26 22
u
13
18
4 4
4
15 15 60 42
3 5
18 189 43 48 80 123 80 50 21 39' 164
6 240 336 366 80 366 80 594 296 2'6 582 213 516 516
24 28 52 33 68 30 30 32 44 32 36 22118 12
6 120 160 154 31154 31 280 132 132 312 135 228 228
"
48
,0
14 38
34 20 30
15 10 10 30 11 20 34 80 5
4
44
31 22
10 40 51 40 10
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5 168 86
10 20
10 16 16
10
21 11 25
10
16
10
16 11
5
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41 80 59
59
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4 10 4
4
6 11 14 26
6
3
4
13 10
6 16 16 16
10
80 30 30 80 55 60 60
11
55 30 30
12
20 40 20
20
40
15
4 12 38
4 21
16
10
6
• 200 66 66 144 135 162 162
1 100 140 180 45 180 45 244 145 145 231 110 251 251
12 5
24 20
4
48 62 13 21 13 21108 60 60 122 50 103 103
13 14
16 31 31 56 50 12 12
15 16
21
a-
-,
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15 20 40 15 30 10 34 31 25 36 15 51 45
16 32 21
n::::r
l>J
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u
u
4
4
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n
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Table 3.
(continued)
ALGEBRA Dl, REPRESENTATIONS Of CLASS 2 l
N
15 23 31 31 39 39 55 47 47 55 63 55 63 87 55 63 63 19 71 11 79 71
-...J 00
95 79 7' 121 79 87 87 81 95 103 103 81 95 103 111 135 95 95 175 103 111 103 Ill 103 103 111 119 119 119 S .P,
I 2 8 6 6 6 2 2 13 2 2 8 13 13 6 6 1 1 6 8 2 2 2 13 26 2 2 6 8 8 26 6 6 2 13 1 2 2 2 1 6 64 448 344 448 240 688 240 448 344 720 448 64 344 688 448 960 120 720 448 720 688 688 440 688 240 '6Q 448 440 440 720 120 344 344 720 960 688 448 688 688 440 880 448 240 240 120 960 960 880 720 120 688 440 0 .s. 22 34 44 46 52 56 58 58 62 64 64 64 66
68 70 70 74 14 76 16 76 78 80 80 82 82 82 84 8' 86 86 86 88 88 88 88 88 90
92
92 92 94 94 !J4 94 94 94 96 9' 98 98 98 LEm
1 1 2 1 1 2 3 6 4 3 1 2 4 S 4 2 5 3 10 17 I 6 5 13 12 11 30 15 20 14 29 4 517~40~~mwn™~W448m™mmmmm~mmmrn~mm~fflmmmmmwmm~~mm~mmm•rnm 64 832 928 824 472 760 768 120 0~ 440 064 ~6 432 256 480 448 COO 768 320 272 312 088 936 088 400 680 032 176 776 000 232 664 496 384 316 440 048 184 840 240 400 096 024 600 056 720 264 880 552 000 144 952 O!M'N
01 30 00 41 1 0 03 4 00 3 00 00 00 00 0 I 30 1 00 0 00 00 00 00 0 0 01 0 01 0 0000001000000001000000100100000000100000102001000000 00100000101000101100103000000010011000102010001000001011001110300000101030000001100000001010200110010010 I O I 1 0 0 I O O 1 2 0 1 1 I O 1 0 3 0 I O I O 1 I O O I O 2 0 I I O 2 0 I I 1 0 1 1 3 0 0 I 1 0 ! 3 0 MEI6HT
4
5
6
1
8
9 10 II 12 13 14 15 16 1l 18 19 20 21 22 23 24 25 26 27 28 29 30 31 32 33 34 35 36 37 38 39 40 41 42 43
44
~
46 47 48
49 50 51 NUMBER
3 2 6 5 5 13 1 8 5 12 1 2 3 2 1 2 1 5 9 1 1 1 2 2 4 2 2 21 21 49 106 84 56 105 294 91 ~ m ~ m m m r n ~ m m m • ~ m m m ~ ~ ~ ™ m m r n m ~ m m ~ m ~ ~ m m ~ r n ~ ~ ~ ~ • • 15 36 29 21 4 10 10
1 1 1 2 4 1 I 3 3 2 1 3 4 3 6 1 1 2 1 1 1 40 114 36 15 85 141 5 6 m m ~ m f f l r u ~ m ~ ~ m m ~ r n f f l m ~ ~ ~ m m m f f l m m ~ m m ~ ~ r n ~ f f i m m m ™ 15 40 15
4
6
1 2 1 1 1 3 1 2 1 3 141 190 390 m 250 469 ~ m ~ m m m ~ ~ m m m m f f i f f l ~ m m ~ m r n m m m m n o m
21 94 1~ 149 60 m
15 36 21
15 12
31 51
29 40 65
1 1 I 2 1 1 IS 114 36 120 150 141 115 234 ~ m ~ m m , o m f f l ~ r u m m ~ m m m r u ~ m m m ~ r n f f l m ~
15
2 1 1 1 36 62 57 28 15 66 72 165 66 110 205 21 291 418 348 303 135 159 304 402 360 157 315 211 753 203 56 481 360 864 804 821 082 981 344 951 800
10
1 I 105 199 175 141 60 225 116 315 615 56 220 145 480 469 335 000 590 125 431 165
10 15 25
6 40 15 46 65 51 50 94 21 105 234 1~ 105
12 21 18 15 24 30 22 69 20 15
4 10
12
36
15
16
24
~
84
I 6 134 168 162 156 67 63 120 186 183 81 123 81 388 556 21 249 205 399 312 438 068 ~o 687 536 021
29 21
4 15 30
14 26 15 21 36 10
65 40
4 4
10 10 10 10 10 10
35 114
64 56 15 75 41 20 47
92 62 51 22 41
15 15 21
14 21 21
25 15
31 40 36 15 15
21
• 115 234
• 150 56
• 225 180 330 90 540
6 121 105 111 160 244 524 191 340 286 145
75 81 66 28 81 66 161 276 21 113 81 220 205 111 494 269 348 232 523
41)
15 12
36 15 120 141
84 91 46 39 25 189 245
10 30 51 21
~
15 50 15 15
10
41)
15
24 16 30 14
4
~
4
15
"
90
35 35
61
60 15 60 60 120 264 66 165 186 120
10
20 21 20 84 104 21 10 10 35
11
21
,o 319
12
~
234
13
. 65
14
42 93 84 96 234 111 162 120 213
15
51 65130 21 54 30 120 94 69 210 mm
46 51
65 21
15 36
21
30 21 24 12 13 24 36 10 15 27 20 78 117
6 56
50 94
• 105 85 145
29
• 40
n "O ....
:::; i,,
...,(1)
..i:,..
4
2
u
1
4
3
8
"" '
'5
3
4 4
' " u u
5
u 5
5 5
4
4
4
'
u
4
" ' ' "
u u
"4 " 5
u
"n
~ ~
u u u u
"u
'
n n n n u u ~ ff u ~
6
ff
GW
~
ff
~
n
"
M
""""n "" " ' u uu u
§
~
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u ~
u
n n
~
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' M
"
~
u
fl
u u
~
~
~
u
~
n
M u u n u n u n
u
1
~
n n n ~ ~m u ff ~ u ~
N
~ C
~
~
~~
~
~
~ ~
~ ~
~
u 4
~
4
4
"
u
4 4
4
" " "u
n
"
~
4
"
~
n
4
~
n
·3 3
1
~
3
4
2
~
2
~
4
ff
n ~
~
2
2
4
~
~ ~
~ ~
~ ~
~
G ~ ~
~
~ ~
~
Table 3.
( continued)
AL6URA 08, REPRESENTATIONS OF CLASS O -----------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------
0
Iv
00 0
8 16 16 24 32 40 24 64 32 32 32 40 40 48 48 56 12 12 80 48 104 56 56 56 144 64 64 56 64 64 80 64 12 80 12 88 88 12 88 '6 112 80 80 80 104 128 120 80 136 152 88 S .P,
I I I 8 6 1 4 2 I 26 10 3 3 I 1 11 I 35 8 35 13 6 43 I 13 26 1 1 4 1 8 8 2 2 101 1 26 1112 1, 120 344 112 224 m 16 128 128 '60 120 168 448 480 688 112 344 224 880 224 152 584 584 16 024 024 120 840 960 840 440 120 oos 344 440 880 168 344 480 m 960 960 688 112 688 520 224 344 8eo o .s.
no
1 21 29 45 51 51 55 55 57 51 51 65 11 11 13 13 11 19 19 81 81 83 83 83 83 85 85 85 81 89 91 93 93 95 91 91 99 9' 99 99 101 101 101 101 101 103 105 105 105 101 101 101 LEVEL I 3 2 4 4 4 10 3 8 6 6 5 3 3 6 6 8 2 8 30 4 3 25 11 5 u e 1 6 6~mru"rn™mm~m~~~m~m™~mm~~mmoommrumm~~mmmm~~rnm~ 1 120 135 820 020 304 925 008 740 435 435 060 312 162 160 800 064 675 255 832 040 880 850 595 595 388 128 128 608 120 400 690 080 500 400 600 816 264 040 132 480 055 544 500 500 625 100 360 600 128 090 150 Dln'N
4 4 0 3 0 4 0 5 I O O O O 4 O O 0 1 0 0 0 0 1 0 0 0 0 0 1 1 0 0 0000000000010000000000000000101100000000002000000000 00000000000000010000200000100000000000000100010020000020000000001000001000000110000000002000000001000001 0 0 0 0 0 0 O O O 2 0 0 0 1 0 O O O O O O O O 2 0 0 2 0 0 0 0 0 1 0 0 I O O O 1 0 0 O 2 0 0 0 O O O O O UEI6HT
4
5
9 10 11 12 13 14 15 16 11 18 1' 20 21 22 23 24 25 26 21 28 29 30 31 32 33 34 35 36 31 38 39 40 41 42 43 44 45 46 41 48 49 50 51 NUflBER
I
6 1,
3
2 14
2
2
'
1
I
1
1
4
4
2
4
5
4
8
3
1
5
4
4
3
2
4
4
6
1
1
1
2
1
4
1
3
2
2
2
1
1
1
1
t
I
I
28 16 56 11 56 28
1 I 35 35 252 416 434 212 512 8'6 315 561 560 288 336 330 815 815
84fflffi™m~fflm~~m~m~~rumm~~~~m™mmm
1' 13 20 15
10 10 81 158 155 '2 180 312 106 205 191 4'9 119 540 355 355
ZB™~mm~mm~~mrum~rum~mmmrnrn™mm~m
21 56 35 42 84 168 63 133 140 154 ,s 210 105 105
ZB~~~m~rurn~mm~~mmm~~mmm~~mmmm
23 45 54 26 52 88 29 58 54 186 32 205 143 143
1 1 4 3 1 1 1 1 1rnrn~rnm~mmmmm~mmmm540~m~ro~~~mm
14 4
1
16 10 11 22 46 18 36 37 55 26 10 40 40
19
I
1
50 50 10,
1
1
7 3
1
1
"m~~mmmm~ffl~~mmffi~rn~m~~mm
24 13 19 25 15 20 15 10 10
10 to 39 30 120 81 85 200 198 240 182 350 165 155 281 180 94 130 130 411 121 456 184 262 205 615
12
10 10
10 16
19 19 10
20 15 65
, 60 81
20
4 4
4
• 213 18 384 304 205 205 330
• 155 119 180
~~llim~~mm~~~mmm~~~
I 1 13 328 891 151 111 450
72 58 58
64 64 101
21 28 21
28 28 28 14 64 35 140 '111 56 336 42 90 196 196 56 35 98 210 190 10 14 64 184 28 21 595
21 21 28
28 28 28 14 64 35 140 '111 56 336 42 90 196 1'6 56 35 98 t,O 210 TO 14 64 784 28 21 595
14 20
• 80 ' ' 120 81 254
56
21
18
4 22
4
4
15 15
'
4 10 4
54 83 52 41 59 59 123 34 132 320 75 58 266
12
40 316 1, 15 250
13
26 20 33 48 25 10 48 22 20 20 20 13 23 80 120 54 36 95
14
120 58 58 135
15
"
30 26 114 21 45 13 lO 30 20 39 15 15 40
5 10 30
18 23
10 11
36 23 34 62 59 88 56 105 24 15 45
n
84133
m 161 204 49 212 800 106 85 639
18 18 36 21 100 65 110 134 120 284 101 213 180 111 148 100 13
51 21 168
35 42 84
24 30 44 26 15
54 52 52
81 23 m
n
:J" I')
-0
...... (D -,
.j:>.
u u
"
u
"
~
u 1
u u
1
4 5
u
n u n n u u
e 9
n
u
~
a
~
~
~
~
~
"
e
M u M
~
u u u
n
~
M
6
u
M
"u "
"
u
"
"u"
u
u
n
n
n
M~
u
u e
"
4
~
"
~ ~
e
~
n
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u
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~
~
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M
$ C
tt
u
0 ~
~
~ ~
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ro ~
~
e e n
M
u
4
4 M
~
8
u
4
a
u 4 4
2
4
4
5 4
4
n ~
~
n n
4 4
~
a
u M
4
4
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M
"4
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3
D
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4
n ~ ~ ~ ~ ~
~ ~ ~ ~
~ ~ ~
~ ~
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Table 3.
( continued}
ALGEBRA 08, REPRESENTATIONS OF CLASS 1
4 12 20
36 20 28 28 36 44 52 68 36 100 44 44 44 52 52 52 60 60 68 16 16 84 60 100 108 68 68 116 68 68 148 84 16 16 196 92 1
I
4
1
1
10
1
I 11
8
4 21 13
1
1
4
1
2 35
1
1
1
8 53
10
7
1
1
92 16 16 84 84 92 84 92 116 92 84 92 108 S .P. 1 53 35
4 17 35 10 13
8
7 107
7
u•mumm~~~mmmummm™~™~m~~™m~m~mmm™~mmmmumm~~~-~m~™™mm•o~. en~~
n
~ u
63 ~
n
~
~ n
n
n
n
n
~
~
~
~ " " " t t tt " " " " " " " n
n
n
n " " ""1n1n31~n5n5n1n1n1n11nn, u~L
I 1 I I I 2 I 2 2 1 2 7 4 6 6 3 3 14 10 8 12 22 10 21 10 10 46 10 6 1 411~~~4644mMttttm•~~mm~m~mm~~mmrnmm~m••rnmmmmm~mmm~~mam 16 560 344 800 368 ~ 192 720 504 800 352 320 688 520 520 280 800 '60 192 800 600 480 504 040 680 648 960 040 760 160 520 240 200 480 680 048 048 040 880 880 600 920 720 000 720 864 600 336 880 000 880 216 DIn 'N
0
0
4
0
0
0
0
1
2
1
0
0
3
O
4
0 0 1 0 0 0000100000000001100000000000000100000000100111011010 0 000 000000100 0 00 00 0 000010000 200 000001000100000 0000010 0 00 00 2 0000 0 0100 0 I 0 02 0 00 00 2 00 0100000000100 00 00 I 02 00 O 0 0 0 0 0 1 0 0 0 0 0 0 0 2 0 0 0 0 1 0 I 0 0 0 0 0 0 0 2 0 0 0 0 0 0 2 0 0 2 0 0 1 0 0 0 0 0 0 2 1 0 0 UE l6HT
4
14
9 10 11 12 13 14 15 16 11 18 t9 20 21 22 23 24 25 26 21 28 2, 30 31 32 33 34 35 36 37
5
1
1
1
1
1
2
1
1
3
3
1 1 8 4 4 6 10 4 9 4 3 18 3 2 84 820 820 001 998 424 265 170 368 660 088 892 515 892 667
2
1
I
1
3 2 1 2 4 2 4 1 1 8 1 1 28 850 850 534 334 '60 185 595 088 320 882 926 925 926 197
10 22 2, 36 31 26 70
so somru~mm~~~mm~rnm~mmm~~mmm
1 1 2 2 1 3 10 10 60 81 8 0 ~ m r n m m ~ ™ ~ ~ ~ m m m m ~ w ~ m m 28 380 380 641 900 '66 355 400 864 290 087 790 895 790 687
6 14
21
4 22
56 35 42 84 57 168
84 133
• 140
• 154 98 210 105 105 28 140 140 294
18 18 58 65 62 161 165 177 103 100 107 185 344 18 85 286 286 82 334 825 38 418 n3 613
18 23 24 44 15 54 52 52 59 110 120 52 58
• 560 567
. 630
. 202
6 182 301 301
60 132 321 32 205 249 249
I 1 156 156 654 414 381 5~ 950 408 918 428 393 111 393 213
~
54 110 26 10 80 80
16 10 11 22 11 46
24 36
31
55 26 10 40 40
so so
50 109
64 125 310
• 220 206
• 265
16
12 24
1' 19
25
15 20 15 IQ 10
10 10 39
30 60 120
• 120 81
85
45
12
19
19
• 60
81
10 10 32 10 21 22
4
""
• 168 315 840
1 1 1 1 1 4 1 386 386 458 083 801 150 901 990 112 185 956 220 956 486 I 1 115 175 540 498 294 420 693 462 630 280 476 960 416 168
14 15 14 63 35 10 21 20 21 35 132 14 15 126 126 14 126 315
4
44 45 46 47 48 49 50 51 NUffBER
2
za~~mrn~m~mm~mm~m•fflmm~~mmmm
4
~
4
21 35 84 111 133 119 98 210
12 t9 19 20 15
38 39 40 41 42
34 63 40 36 36 30 30
16 15 19 10 10 11 20 44
10 41
20 15
41 10 48 115
12 104 104
4
50 2e3 155 188 2so ~5 no 480 206 160 100 160 156
7 10 10
64 64 231 188 135 190 328 189 314 145 196 m
14
10 196 91
21
14
21 15
14 35
14
15 21
14 35
21 42 49
11 14
12 12
19 36
4 22 30 30
21 49 42
4
11 12
28
28 63 84 35 50 84 84 11 35 105 350 91 21
13
28
28 63 84 35 50 84 84 11 35 91 350 105 21
14
18 18 96 69 63 82 142 18 158 65 80 299 BO 51
15
n::T i,.,
"O
...
Fil"
.i::,.
4
4
8
4 4
2:
18
6.
4 5
2'
2
2
t
4
3
2
16
11
13
4
T
' 4
24
21 42 100
.,
• 110
24 30 15 20 34 34 36 15 40 125 40 11
12 12
10
4
4
4
12 65
3, 24 31 31 54 33 80 23 32 102 32 30
18 18 36
4 22 15 15
8
'
18 36
38 23
34
10 24
16 15
45
4
10
20
10
1
10
'
1 12 14 1T
10
18 1, 20
11
21 22
23
18
15
16 39 16
s: C
;:;-
-o· o· ;:;: -< -I
23
l'J CS"
24
CD V>
25 26 21
4
10 15 15 10
4
28 2, 30
4
31 4
13
32 33 34
4
35 36
31 38 3'
4
40 41 42 43
44 45 46 41
48
"'50 51
N
00
w
Table 3.
(continued)
ALGEBRA 118, REPRESENTIITIONS OF CLASS 2
8 16 24 32 32 40 56 40 48 48 56 64 56 88 64 12 56 64 12 64 80 96 12 128 88 12 80 80 80 80 88 88 104 104 88 112 88 136 '6 96 104 96 116 104 112 104 112 96 112 104 120 104 S .P. I 3 1 1 1 1 8 1 21 3 1 3 1 1 35 21 21 35 1 1 21 1 1 53 1 1 35 11 43 35 3 21 35 1 53 1 21 21 3 101 1 1 1 21 35 11 21 8 43 11 128 024 584 024 168 168 024 '60 168 504 584 168 584 024 024 128 840 504 504 840 168 168 504 024 024 160 168 168 840 680 008 840 584 504 840 168 160 168 504 504 584 520 024 168 024 504 840 680 504 960 008 680 0 .s. 2, 43 55 51 65 69 11 13 1, 19 81 83 83 85 85 85 81 91 93 93 95 91 91 99 99 9' 99 101 101 105 105 101 101 101 10, 109 109 111 111 111111 113 113 113 113 115 115 115 111 111 119 119 LE~EL 2 3 4 4 4 4 6 1 2 11 5 1 14 24 2, 30 11 22 33 23 31 16 45 36 15 104 4 43 18 15 12 83 81 51 164 132 1u~~~nmmrn~m~mfflmmmm~~mmafflm~~mmmm~rnrum~rnmmmm~~mmm~mmmm 128 920 312 360 320 ~ 040 280 144 208 1'G 560 920 680 296 040 136 120 904 960 240 116 360 384 344 ~ 480 032 152 120 960 040 000 680 144 920 000 800 040 680 360 880 280 520 624 616 120 120 040 160 000 840 Olff'N
0 0 0 3 1 0 2 0 0 0 I 2 0 4 0 0 2 0 0 2 0 4 2 0 1 6 2 0 2 1 3 5 1 1 0 2 0 0 0 2 0 0 0 0 1 0 3 0 1 0 0 0 2 1 I 0 1 1 0 0 0 0 0 1 0 0 0 0 1 0 2 1 0 0 0 1 0 1 0 0 0 I 0 0 1 0 0 0 0 0 0 0 0 0 0 2 0 0 0 0 1 0 0 0 0 0 0 0 0 1 1 0 0 0 0 0 1 0 2 0 0 0 0 0 0 0 0 0 1 0 0 0 0 0 1 0 0 0 0 0 0 0 0 1 0 1 0 0 0 0 0 0 0 0 0 0 1 0 0 0 0 0 0 0 0 0 0 1 0 0 0 0 0 0 0 0 0 1 0 1 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 1 0 0 0 0 10001010000000100010101011101030000000100000010000102010100010001010001011101110300010103000001000100001 0 1 0 0 1 1 1 0 1 0 0 0 0 0 2 0 1 1 1 0 1 1 1 1 3 0 1 0 0 1 0 1 0 0 1 Q 0 0 0 2 0 1 0 2 0 1 1 0 1 0 1 1 UEl6HT
--------------------------1
2
4
,
------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------
5
9 316151818 10 31 24 3 10 8 3 24 1 1 2 1 1 1 2 4 1 2 4 1 9 8 3 6 8 28 28 16 161 84 154 239 518 m 514 280 210 232 t t r n m m m ~ m m ~ m r n r u r u ~ m ~ m m m ~ m m m m m ~ m m ™ m r u r u r n ~ m m m
' '
4 1 8 1 8 8 5 15 12 1 I 1 2 3 4 3 1 2 4 2· 4 I 5 4 1 11 21 49 28 49 84 182 133 211 105 84 91 3 6 ~ r u ™ m m m m ~ m ~ m ~ m m m m m m m f f l m ~ m m m m ~ ~ m m r u m m m m m
12
2 3 3 4 4 2 1 6 1 1 1 2 2 1 5 1 1 1 15 29 ST 43 TO 40 28 36 15 163 246 323 318 301 213 435 9 4 m m m ~ m r u m m m m r u m r n ~ ~ m m f f l m r n m m m ~ w r n ~ r n 21 21 49 4 10 15 11 18 .15 12
2 2 1 2 1 4 2 1 2 49 106 182 84 189 211 105 84 3 6 m t t r u ~ m ~ m m ~ m m m ~ l l i m m r u m m m m m m m r n m m
28
15
1 2 1 1 1 2 1 1 3 3 1 55 83 108 123 101 91 185 28 ~ m m m ~ r u m m ~ m m ~ m ~ ™ m ~ ~ 84 108 463 103 611 053 891 224 424 190
4
1 2 1 1 15 36 51 2, 64 10 40 28 15 114 36 85 163 234 381 318 168 323 360 351 330 213 435 306 126 233 84 391 162 921 922 829 118 540 182 245 21
21 49
16 23 30 46 21 24 11 4
15
• 49
28
4 33
4 10 15 10 11 18 15
• 106 84 56 182
• 189
• 294 84 tt 36 216
~
• 336
• 994
1 1 1 34 108 81 68 130 218 248 328 90 114 353 114 461 112 533 465 204 022 28 531 232 104 666 '51 118 569 408 514 18 33 36 20 40 112 T2 130 25 50 m 6 40
12 12
is
12
36 29 21 51
15 12
19
10 38 16 45
14
15
10 40
64
5 10 44
95 21
21
l
48 205 30 239 231 ,a 3,0
1 264 115 263 250 419 288 249 515 114
n:::::r
31 55 94 134 123 58 108 149 122 145 91 185 141 60 496 28 111 15 380 362 380 456 250 818 605 15 ff 29 12 29 43 51 65 85 40 10 40 36 15 228 28 36 15 205 163 150 281 115 528 225
15
'
• 211 105
14
• 114 28 36 15 85 163
5 108 108 58125
35
42 63 21 35
• 381
1 121 64 81 80 181 n 104 155 362 28
49
• 141
4' 10 28
• 106
21 20 63 21 35 35 182
10 11 12 13 14
i:>J
.......,
"O
(1)
.i:,.
12 4
4 10 20 10 11 15 25 23 15 18 15 15 4 4
10 10 15
12
15
18 15
1l 21
18
15 16
15
n
40
15
31 55
15 15
4
98 50 185 105
17
51
18
12 16
15
19
• 134
49
45 92 134
19
36
12
20
15
12 10
55 65
36 18 42 40 69
15
14
28
80
4 33 30 15 51
12
12
64
81 34 140 130 165 165 110 314 291
28 28
21 21 28
20
16 36 38 4'i 16 30 51 33 62 24 11 66 28 183
10 10 24 11
36
21
15 20 64
22
14
14
47
25
14
26
/3
4
4
24
23 16 26 28 11 53
15
4
!2
4 10 16
15 10 30 15
21
15
38
28
11 21
19
10
20
30
4
4
~ C
!:!". ""Cl
~-
rt-
-< -I PJ C,
;;, V,
31
4
32 4
10
33
34 35 36
31 38 4
39 40 41 42 43 44 45
4'i 41 48
49
so 51
N
00 V,
Table 3. AL6EBRA
o,,
(continued) REPRESENTATIONS OF CLASS 0
N
00 C'I
16 16 24 32 40 64 24 32 32 40 48 48 40 48 48 56 12 12 80 104 48 144 56 56 56 64 64 64 80 12 64 80 12 12 88 88 12 '6 112 12 80 80 104 88 80 128 '6 ,6 120 136 S.P. 22 522012 832224 2 80 3240642188020241611221212241'3288018184324 224 1 144 18 016 016 144 288 18 316 304 160 0'6 612 064 256 304 304 032 144 016 288 288 640 18 256 320 512 016 432 640 160 1'2 280 0,6 504 504 016 1'2 536 016 064 640 432 432 032 256 032 144 304 304 032 288 0 .s, 1 31 33 53 5, 61 63 65 61
n
11
n
85 85 81 8,
a,
e, ,1 ,1 ,3 ,5 ,1 ,1 " 103 103 105 105 101 10, 111 113 113 115 115 115 115 111 111 111 11, 11, 11, t1' 1i, 121 121 121 121 121 12a LEVEL
u
u
2 3 5 2 6 u 6 13 11 u 13 13 u n 43 ~ u 33 33 n M 5 20 20 22 11 311 8M 5 u 4 3 r n r u r u m r u m m m m r u m m r n M f f l m m m ~ ~ r n m ~ r n r u r u m ~ f f l ~ m m ~ ~ ~ ~ m ~ m m ~ ™ 1 153 110 060 415 550 212 814 564 158 600 328 '55 065 510 822 822 632 800 800 506 116 800 '62 206 240 140 436 ,48 352 425 225 '50 138 566 566 260 560 800 224 550 260 600 600 350 864 612 424 050 050 0,6 240 Dlff'N 0 1 2 4 0 1 0 1 2 4 0 1 0 0 0 1 0 4 6 2 1 4 0 0 1 0 0 0 0 1 0 0 0 1 3 1 0 0 1 1 0 0 2 0 0 0 0 0 1 0 t 1 0 0 0 0 0 0 0 0 0 0 0 0 1 1 0 0 0 0 1 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 1 0 0 1 0 0 0 0 0 0 0 0 0 0 0 1 0 0 0 0 0 0 0 0 0 0 0 0 0 1 0 1 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 1 0 0 0 0 0 0 Q 0 0 0 00000000000000000010000000000120000000000000000000001000100000000100200000000100000020000001000020000000 0 0 0 0 0 0 0 0 0 1 0 0 0 0 0 0 2 0 0 0 0 0 0 0 0 0 1 0 1 0 0 0 0 0 0 2 0 0 0 0 0 0 0 2 0 0 0 0 0 2 0 0 UEI6HT
0
1
4
36
"
5
6
l
8
,
10 11 12 13 14 15 16 11 18 1' 20 21 22 23 24 25 26 21 28 2, 30 31 32 33 34 35 36 31 38 3, 40 41 42 43 44 45 46 47 48 4, 50 51 NUftBER
7 6 6 6 12 21) 8 5 11 13 13 11 10 5 2 7 7 10 5 1 2 2 3 4 1 4 8 4 l2 100 36 84 126 384 120 411 1n r,9 504 504 344 464 848 828 4,2 352 120 430 456 032 812, 116 064 413 441 1,8 8'6 216 216 229 816 160 4'6 278 844 440 440 284 800 760 5,2 140 140 1s2 280
' '
22 15 23 16 2
3
3 3 4 2 1 1 1 3 1 3 3 3 2 2 2 5 8 3 2 5 5 5 4 4 2 21 35 112 216 125 245 266 115 11S 420 141 214 261 156 840 3 6 m m ~ r n ~ m ~ r n ~ ~ r n r n ~ ~ ~ ™ m ~ ~ ~ ~ m - m m m m ~ 1 t 2 1 1 1 1 1 2 3 2 1 1 2 2 2 2 1 1 56 35 35 224 84 116 184 128 252 3 6 l l i r u f f l r u ~ ™ m m ~ m m m ™ ~ m ~ ™ ~ ~ ~ ™ m m m ~ ~ m ~
28 16 56 m
10 28 54 31 62 85 60 60 104 34 68 63 31 282
1 1 1 1 3 1 2 2 2 1 1 1 1 1 1 1 1 1 8rufflmm~~rn™rn~mrnrum*rn~~mmrummm~~-~
1' 13 26 ts 10 10 54 21 42 43 30 81
1 8 102 158 155 n m ~ m f f l f f l r n m m m m m m m m ~ ~ m r u w m m m m m
28 15 22 2, 23 21
21 56 35 42 ~ m m m ~ m m m m ~ ~ ™ ~ m m m ~ m m m m r n r u m
14
7 22 22 23
~
21
10 26 21 21 16
3
4
4
10
6 4
4
84 112 111
• 133
• 11' 34, 210 244 245 • 16
• 28
16
• 56 112
• 140 140 311 266
• 105 115 115 518 274
• 11 56
28 35 35 224 112
85
1 1 1 ,3 12, 222 6 1 ~ r u m m f f l ~ ~ ~ m ~ m m r u m ~ ~ f f l m m 81 602 602 3'2 11,
20
21 28 84 14 '2 64 ~ ~ m 5 6 r n r n 4 2 ~ ~ 5 6 ~ m ~ ~ ~ m ~ 14 260 260 64 28
4 23
28 45 54 26 64 124 80~™~™™mmmrnmmmm™mm 58 216 216 252 125 1
12
1'
10
n::r
l'J
62 10, 100 100 146 85 64 40 60 60 156 88
11
36 15 31 10 10 3' 51 30 S7 26 3, 20 20 86 15 30 27 10 10 ,4 63
12
....Ft
21 60 60 130 68
13
~
16 10 11 10 44 28 75 55 11 40 40 61 U6 140 6
4
10 22 28 2,
36
22 24 15 30 65
2fj
20
31
,o
"
10 62 62
50 50
"
85
58 58 21 45 162 30 20 101 142 142 40 ,8 53 28 21
• 48
.286456
28 14
,
85 85 40 19
14
36 36
15
"'O
n
H
u
tt
~
~
tt
~
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u u u u u H n
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n u n
u
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9: ~
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ro ~
u n
w 4
~
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4
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~
4
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a 4
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a ~
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~ ~
~ ~ ~
~
ft
N 00 ~
Table 3.
( continued}
ALGEBRA 09, REPRESENTATIONS Of CLASS 1 ----------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------4 12 20
36
20
28 28
36 36
36
44 52 68 10Q 36 44 44 52
52
60 52
68 16 60
60 16 60
68
68
84 100 108 60 116 148 68 68 196
68
84 16 16 84 84 92 16 92
84 84 92 116 84 S .P.
18 612 288
4 8 4 2 2 32 18 40 20 8 24 96 2 8 5 9 2 32 2 2 4 2 101 20 161 129 32 120 129 40 8 18 1£1 80 32 32 24 20 32 18 032 064 608 256 256 016 016 288 288 18 256 432 320 160 064 192 168 016 064 216 216 016 256 304 304 032 288 016 520 288 288 160 280 18 024 256 960 024 320 064 432 280 640 256 256 H2 160 256 O.S.
11 43
49
41
61
69 11 13 13 13 15 11 1' 81
83 89
91
93
95
99 101 101 101 103 103 103 103 105 105 105 101 101 109 109 111 113 113 113 115 115 11' 119 121 121 121 123 123 125 125 125 125 121 LEVEL
1 3 3 2 2 2 2 2 4 3 3 4 2 2 9 2 I 9 24 24 15 41 60 39 28 31 51 69 42 42 66 31 50 1 1 8 45 31 24 24 61 90 84 18 25 293 101 930 108 116 281 11' 211 188 956 956 414 311 401 401 558 154 386 302 494 364 883 066 31' 186 049 831 648 613 139 209 395 221 401 401 853 461 605 18 816 920 122 568 696 824 310 310 830 288 150 336 194 160 200 240 536 288 850 100 012 800 096 096 010 296 190 190 116 240 800 400 206 352 800 900 110 240 216 592 588 920 160 600 160 298 502 502 248 800 056
I
3
4
0 0 0
0 1 0
0
0
Dln'N
0000000000000010000000000000000010001001000000000001 0 0 0 0 0 0 0 0 0 0 0 0 0 1 2 0 0 0 0 0 0 Q O O O O O O O O 1 0 0 0 0 0 0 0 0 0 0 1 0 0 0 0 2 0 0 0 0 0 0 1 2 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 ·O 1 0 0 0 0 0 I O O O O O 1 0 0 1 0 0 2 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 2 0 0 0 0 0 0 1 0 0 0 0 0 0 0 0 2 0 0 0 2 0 0 0 0 0 0 0 0 0 1 0 0 1 0 0 1 0 0 0 2 0 0 OU£16HT
9 10 11 12 13
16
28 112 56
35
14 15 16 11 18
34 42 36 30 14 22 22
23
24 25 26
21
28
29
30 31 32
33 34 35 36 31 38 39 40
41 42
43
44 45
46 41
48
49 50 51 NUflBER
23
1 3 3 1 6 1 4 3 3 5 1 4 4 1 3 4 1 8 ~ r n ~ m ~ m m r n ~ l l i l l i f f i ~ m m f f l r u m m m m m m 36 138 ,,0 060 089 122 325 930 441 134 044 044 206 118 168 21
1 3 1 1 1 1 2 2 2 1 1 2 3 I 35 B 4 m m ~ m ™ ~ ~ ~ m ~ m m m ~ m m m m m ~ 36 029 903 144 639 248 602 645 016 948 410 410 145 164 904
16 4
22
2 2 1 1 1 1 1 2 1 1 2 I 1 3 1 3 8 1 4 15 16 11 8 8 12 18 8 8 11 l 10 35 141 116 156 128 3 6 m m m m ~ m ~ ~ ~ ~ • m m ~ ~ m r n m m m m m m m ™ m m 5 4 8 m ~ ~ ~ m m m ~ ™ m
26 15 10 10
4
19 20 21
4
16
28 28
56 112
11 56
35 35 224 112 116
64 12 80 16 200 261 124 120 20Q 200 128 290 216 216 220 10 22 28
29
90 10 62 62
22 14 15
14 35 90
44 521 290
85
1 530 238 102 22~ 548 3T8 694 565 882 813"~13 198 350 860
,e
85
28
28
36 36
21
28
36
28 36
10 10 40 15 10 20 15 10
35 14 15 180 14
36 10 T4 42
6 168 420
48
42 105
. 216
56 162 504 120 84 288 384 189 318 351 168 10 336
48
42 105
• 216
56 162 504 120 84 288 384 189 351 318 168 10 336
42 60
so
30 80 162
10 54 28 42
43 30
81
28 22 22
29 23
21
14
22
22
8 120 102 412 310 360 211 190 304 624 225 225 684 292 328
18 11 26
21
21
20
10 58 10
21
23
21
62 ISO
• 444
• 112 m
10
35
• 168
. 168 112
11
35
84
• 112
12
28
13
• 102 158 155 180 ,2 1'0
16 15 22 10 10 18
• 896 896
1 I 1 8 410 308 912 035 UQO 569 635 848 390 650 650 331 618 816
14
4
1 • 344
60 130 61 68 185 10 31 200 488
20 18 18 21
13 26
• 336 416 434 512 212 560
1 2 2 1 1 1 2 2 1 1 2 1 2 8 296 136 220 969 T30 214 668 324 844 816 816 620 150 064
1
96
36
60
36
21
,2 100 548
• 252
10 85
21
12 38 15 32 26 1,
so so
• 184 128
1 161
56
35 84 42
93 2,4 35' 240 110 241 344 412 2e, 286 3n 180 334
14
t-0 00 00
4
4
4
4
4
11
14
14 35
18
4 24 46 4
4
4
35 126
15 16
10
4
4
4
21
28
23 10 10
63 154 50
16
35 92132
4S 54 52 26
64
84 126 126
88 80
• 124
17
48 20 38 38 10 40 66 30 30 97 28 54 16
10
1T
44 28 120 108 102 18 64 120 116 % 96 194 80 132
63
10
24 38
44
22 11 10
46 28
20 15 22 44 34 41
41
36
15
48
18
$
19
!:!.
20
C "CJ ()
14
14
21
15
14
14
15
21
-l
14
23
0-
14
24
V,
12 19 10
22
24
16
.....
'
26
n 12 4
19
28 29
11
9
30
31 32 4
3
4
5
33 34 35 36
4
2
4
4
31
38 4
3'
40
41 )
"O
10
'8
U 135 28 92 ,2 141 39 190 28 56
,8 148 56 110 35 35 56 52
12
10
10
10 40 21 40 41
10 31
31
41
32
15
78 40 60
61
42
16
10 20 10 40 31 31
41
32
1, 38
28
2, to
36
35
75 1s, 10
20 10 31
• 156 156 200 31
17
18 32
1,
11
54
66 55 55
20
11
54
55 66 55
21
10
11
10 24 15 30 26 40 21 45 4' 30 20 40 13 12 4
11 21
16 10 23
15
10 10 15
22
£?: .....
-
4
10
16
10 26
21
35 92 15 88 64 140 106 40 186 114 411 160 418 20
4
24 25
10 10 25 30 15 26 19 60 30 10 110 109 155 130 115
26
36 11 55 44 18 12 69 183 63 116
21
4
12 35
4
14
12 12 19 32 18 24 21
12
18
10
4
18 15 24 11
69 18 120
28
24 22 80 20 93
29
4 39
44 68 49 16 14
4 13 13 24 14 15 14
4 10
4
4
4 4
31
4()
32
45
34
35 30
30
15
33
10
35 15
36
12 14 28 15 32
31
10 10 to 10
38
4
10
4 10
39 14
40
41 4
42
4
43 44 45 46
41 48 49 50 51
w -..J
w ..... 00
Table 3.
(continued)
lliEIRA El, REPRESEITIITIOIS Of CLISS O 0
2
4
10 12 14 18 14 16 16 18 20 20 22 22 24 26 32 24 26 2' 28 28 30 34 30 32 32 36 32 34 36 34 38 42 5G 36 38 38 40 38 40 44 40 42 42 44 44 46 S.P.
1 1 n 30 60 13 " 2 138 111 30 241 111 138 6 13 483 30 483 138 6114 362 30 483 n 138 2 '61 111 111 ,n 13 13 451 181 483 30 '61 ,n 138 20, 483 814 138 181 m 2 6 1 240 160 120 240 280 240 480 440 240 120 160 240 440 240 ,20 440 240 120 440 240 840 240 8«1 240 100 880 240 840 280 240 160 '80 440 440 '80 440 440 240 520 440 840 240 '80 680 240 600 840 400 240 440 880 0 .s . 1 5,
,3 115 m 131 151 m m 115 183 185 1,5 211 20, 221 221 22, 22' 231 233 241 243 251 253 m 265 zn 211 213 215 211 m 283 285 281 281 2e, 2,i m 2" 2" 301 305 30, m 313 m m 321 m
323 LE~EL
1 2 2 4 8 3 6 4 8 2 20 21 23 45 11 12 2 85 66 83 63 124 234 110 220 2'1 514 223 355 508 2416 42'U~W™~~m~m~mmmrnmmmrnmm~mmrum~~m•m~mmmmmrnrnm 33021Wm8~mmm~rn~m~~~~wm~~rnrnmmmmmmm~mru~~mm~m~*mmm•~mmm 1 248 815 380 000 250 241 240 000 125 000 384 008 625 000 210 500 '16 000 815 000 000 000 000 860 125 000 UO 264 250 000 000 '52 000 045 110 000 '60 280 000 000 2SO 120 000 000 000 625 840 368 000 000 150 DIn 'I
01 00 0000201 001 020 10 00010000202 00 1 00 00 0 0 0 0 0 0 1 0 0 0 0 0 0 0 0 0 0 1 0 0 0 0 0 0 0 0 1 0 0 0 0 0 0 0 0 1 0 1 0 0 00 010 0 O 0 00000000001000000000000010000000001000000000001010000000012000000000001000000000000000101010001000100000 0 0 0 0 0 0 O O O O O O O O O 1 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 1 0 0 0 0 0 0 0 0 O O O O O 1 O O O O 0 000000010 00000001000000001000 00001100000000100001 O O 0 0001000010000100002100010010000001002101100010000021 0 1 0 0 2 0 1 0 1 3 0 0 1 0 2 0 1 0 0 2 4 1 1 0 2 0 1 3 0 0 0 0 1 0 2 1 1 3 5 0 0 2 2 0 1 3 0 0 1 0 0 2 UEffiHT
4
5
1
8
,
10 11 12 13 14 15 16 1l 18 1' 20 21 22 23 24 25 26 21 28 2, 30 31 32 33 34 35 36 3T 38 3, 40 41 42 43 44 45 46 41 48 4, 50 51 NUnB£R
1 1 1 1 3 3 3 1 2 1 11 8 10 1 15 26 11 24 26 51 21 34 41 1 2 4 1 , 418433T"rn™~"25ffi243~mmrnmmmmffl~0ru™mmmmm~~mmmm~™mmm "~mmffl~•mm~~~~m~rumrumrn~440mm~~*m~mrummrnm~~m~m~~m240m~™~~m 1 1 1 3 1 6 4 5 4 8 14 6 13 15 32 12 1, 26 11 2 1111u21H~M4511rnmmzn~mm~mm~~mm~~~mrnrnrum™mm~mmwmm 1~~mm~rn~mfflMmmw~840m•mmmmmmru~™mmrum~rnmm~mrummm~mmrum~mm~ 1 3 2 3 2 4 1 3 1 8 1l 6 10 14 2 16uu22u205n»mmmfflm™rnfflrnmrummm~~rurumm~mm™mrnmmm ~rn~~mrum~mfflmm840rnrummru™mmm~~mfflru~m~mmru~mmm~™m~m~™~rn~ 111124134,358 1 2 2 4 11 9 6 8 2 32 23 62 58 124 215 94 16' ,5 200 65 460 461 505 '10 36' 256 5G 105 2,4 u, 20, 342 263 ,41 ,20 532 132 101 890 232 ,~mrnttrnrn~~mm~w*m~mmmm~~~~mmmmm~mm*~rurnrum~mmru~~™~~
121225234 I 1 4 3 2 3 1 13 10 Z1 26 Sl !00 45 18 45 '6 31 223 221 252 482 184 131 26 869 665 833 627 211 244 031 076 434 232 004 189 491 121~~n~m~rum~~rnm~mmrnrummmrnmm~m~mmmm~rn~~m™m™~~~~~m 121215234 1 143 z 3 u"2126~10C441845'6nmmm~mrn26on~mmm~mrn~m•m& 1~48n~~™wme~m™mm~m™mmmm~mmm~mm~mmmmrummmm~mmmm•~
6 12
I 112112 1 I 1 1 5 4 11 11 25 45 20 3' 21 45 15 106 109 122 234 90 64 13 435 335 420 319 621 160 542 083 289 768 069 699 408 6 22 11 104 338 323 105 111 508 021 442 414 666 214 961 461 503 656 155 m 442 360 181 264 559 302 813 444 916 29' 0,1 927 363 11e n6 381 296 253 469 896 636 441 100 1
1
2 1 4 4 10 20 9 16 9 20 7 49 51 58 111 43 31 6 213 166 201 159 312 589 218 554 671 441 S61 891 171 4~rnm™m~mmmmmmmm~m~mmm~~m~m~mmmmm~mmmm~rn~m 1 I 4 8 4 6 4 9 3 21 23 21 51 20 15 3 101 79 100 ll 152 294 142 271 341 737 287 460 666 21 34 11 216 171 139 219 n r u ~ m m ~ m m ~ m ~ m m m m ~ m ~ ~ r n m ~ m m r n m ~ m ~ m m ~ 21 6
28
21
! 1 3 4 6 9 4 4 1 20 16 22 11 33 69 36 63 80 182 69 117 179 56 ~ n ~ m m m m m ~ m ™ ™ m ~ m m m ~ m ~ m ~ m ~ r n ~ m f f i ~ r n ~ m
I I 4 8 3 7 4 9 3 21 23 26 51 20 14 3 101 80 100 17 153 291 140 218 343 13' 289 459 660 21 n 203 190 121 1n ~ m ~ m m ™ ~ m ~ m m m ~ ~ m r n m m ~ m m m r n m m m ~ m m m ~ m
n
1 3 1 2 1 3 1 9 10 11 23 9 6 1 41 31 41 37 73 142 '8 131 172 369 141 232 336 21 56 64 42 Sl n m m m m m m ™ ™ ~ m ~ ~ ~ m m m m m ~ ~ m ~ m ~ m m m ~ ~ m r n
u
1 3 I 2 1 3 1 , 10 12 23 9 1 1 41 31 41 31 13 142 6' 136 111 369 146 232 338 41 63 n m m m m m • ~ m m m m m ~ m r n w * ~ m m m m m m m m ~ 0 m ~ m
u
5 16 16 11 11
I 1 I 3 4 5 10 4 3 21 11 21 11 34 U 33 65 84 181 12 115 168 HM~mrumrnmmrum~~mmmmmmmruwmrumm~~mmrn
u
12
1 1 2 4 1 1 9 1 9 1 15 31 16 29 39 87 33 56 85 22 11 75 104 212 528 323 3,0 248 544 203 584 877 412 036 911 632 414 206 471 896 849 184 m 832 943 111 128 m m 0a1
14
1 1 2 4 I I 9 1 9 7 15 31 15 30 40 86 3S 5S 82 1~n18~mmm~mmm~rnmmmm~mrnmmmm~mru~mrnm
e
1 3 3 4 3 6 14 1 13 18 40 16 26 40 4U~~mm~'6mn™mmmmmrnmm~mmm~mmm~mm
u
1 3 3 4 3 6 13 7 14 19 40 16 26 38 1 21 21 n 154 11 161 106 244 ,s sa1 686 813 674 658 518 126 954 332 039 339 e1a 859 014 126 015 349 842 1n ,11
11
21 63 56
4
2
w
I',)
0
Table 3.
(continued}
ALGEBRA E8, REPRESENTATIONS OF CLASS 0
0 1 0000000000100000000001000000001000000001010000010000 00000000001000000000000010000000001000000000001010000000011000000000001000000000000000101010001000100000 0000000000000001000000000000000010000000000000100000 0000000100000000100000000100000001100000000100001000 0001000010000100001100010010000001002101100010000021 0 I O O 2 0 1 0 1 3 0 0 1 0 1 0 1 0 0 1 4 1 1 0 1 0 1 3 0 0 0 0 1 0 1 1 1 3 5 0 0 2 2 0 1 3 0 0 1 0 0 1 UUGHT
9 10 11
11
13
14 15 16 17
18
19 10 11 11
13
14 25
20
26
17
28
54 34 45 21
29
30
31
32
33
34 35
48 12 12
40
41
42
43
44 45
46
41
48
49
50
51
NUMBER
19
I • 504 512
20
I 1 12 18 1 1 1 2 6 3 6 8 18 56 519 344 613 394 856 036 114 142 534 215 620 060 303
21
96 224 81 11
1 1 1 1 3 1 3 5 8 21 607 563 624 563 148 423 264 661 835 948 553 313 001
22
86 108 118 91 18
1 1 1 2 3 7 3 5 8 21 601 522 646 551 156 493 334 603 149 911 391 336 190
13
35
63
1 1 3 1 2 4 11 196 161 180 124 399 099 129 910 344 _430 111 444 206
14
72 26 11
1 1 3 1 2 3 6 116 199 130 207 442 961 516 014 610 334 500 156 465
25
so 34 18
15
39
l8
11 111 216 111 184 219
15
38
1 1 2 3 8 2 5 10 14 531 432 806 630 999 850 962 145 024 222 101 815 260
34
21
37
1 1 1 1 2 6 3 5 8 18 1 12 18 36 64 24 206 273 351 603 302 164 14 551 196 697 413 846 150 312 991 344 350 354 214 846
27 21
36
J2 206 255 318 620 162 218
10 32 17 60 11
11 56
24 10 66 11 4 18
80
63
14 30 10
16
56
56
16 16
11 12
29
11
66
63
• 225 336
. 588
n::r s,J
'"O
1 1 57 88 14 149 397 254 365 510 364 526 984 657
26
..... ...,(1)
. 190 528
21
+'-
22 11
75 104
. 212
4
22
4
11
69
76
1 1 73 160 347 186 420 664 328 631 903 395
18
3::
20
21
21
28
56 154 266 498 152 J36 515
29
!:!'.
C
21
28
21
27
56 112 49 106
56 161 266 497 181 343 519
20
18
25
22
31
81 139 131 510 210 369 609
32
-< -I
44 28 44 80 180 75 143 237
33
er
51 131
4 16
15
;:;·
57
21 4
30
"O
22
18
36
11
50
35
56
64 42
70
67170
34
90 177
91 116 104
35
21
54
36
21
31
20
r+
~
(I) V,
38 4
10
30
56
31
42
68
5 10
16
16
11
11
40
20
48
41
12
42
18
43
11
44
15
11
10
15
10
11
39
45 4
46 47
48 49
so 51
w N
-'
Table 3.
(continued) w
ALGEBRA EB, REPRESENTATIONS OF CLASS 0
5? 44 % % % 50 48 48 50
N N
48 54 54 50 56 62 12 50 52 52 52 54 56 58 54 54 56 56 56 58 58 64 64 56 58 58 60 60 62
2 1 30 419 138 961 451 30 60 961 362 903
6 181 '67
13 13
419 604 903 209 362 483 30 451 903 138 69 483 138 967
66 14 60 62 62 62 62 64 66 64 64 68 66 68 S .P.
4 1 4 2 138 838 814 903 814 961 362 362 30 838 483 419 967 903 138 181 961 903
30 181 967
™m™~~™~~m~rn~~~~™m~~mm~~~~~m~~m~~~~~~mmm~~~~~~™~~~™~rno~.
325 329 331 333 335 335 337 331 341 343 343 343 345 345 341 349 351 353 355 351 351 357 359 363 363 365 365 365 367 361 369 369 371 315 311 377 379 319 381 383 385 387 381 389 389 389 391 391 393 393 395 395 LE~EL l 1 1 1 3 I 3 4 3 8 4 6 3 3 7 13 5 3 8 4 12 4 26 z, 33 33 28 26 19 4 56 36 58 46 78 36 28 18 107 28 51 51 mm~mmmm~mmm~m~mom~~mmm~mmm™mmmm~rnwm~rn~m~~m~rnm~™™m~•m
m~mm~mllim~mmrum~mrnmru~~mmmm~mmfflmrnmm~mmmm~~mmmmmm345m~m•"'m ~~m™mmmmmm~mmmmm~m~mmm~mrn~~~mmmmm~~m™~~m~~~m~~~~mill~™
500
ono
500
ooo
840
ooo
250 815 920 920
ooo ooo ooo
450 500 200 646
ooo ooo
500 012
ooo
750 125 750 120 120 800
ooo ooo ooo ooo ooo
2so
ooo ooo ooo oao ooo ooo ooo
250
m
184 944
ooo ooo ooo ooo
208 192
ooo 01n'N
10 I 3000 0 02 202 00 040110 1 1000010310300 0001000000000000101001000020100000100000000010010000
O 00120 00 0 00 00 0 000010000010 00 000 00 0 0000100 000 00 011020001000100010 00 00 IO O O t O 00 0000012 O O 100 O 02 0 O O 00 00 O O O O O O
0000100100000000000100000000000010000000000100001001 0000002010010000110000001000000001010000000100000020 0000100011300210001010000002010001000210110010101000 4 1 1 1 0 1 0 2 1 0 0 3 2 2 i 6 0 0 1 0 1 3 3 0 1 0 0 0 0 2 0 4 1 0 I 2 1 1 3 5 0 0 2 0 0 2 0 2 I 2 1 3 UEIGHT
52 53 54 55 56 51 58 59 60
61 62 63 64 65
66 6l 68 69 '111
71 72 73 74 75 76 11 78 7' 80 81 82 83 84 85 86 8l 88 89 90 91 9Z 93 94 95
111111
96 97 98 99 100 101 101 103 NUMBER
212231134111
net~~mu~mm™~nru~u 1 m m m r u m ~ m ~ m r u m ~ ~ m ~ ~ m m m ~ m m m r n r n r n m ~ m ~ m m ~ m • m rnm~~mmrn~mmmrnmm~™mmrn~~ffi~~mmmmmmmm~rnmmm~~~mmm~~rnmmrnm~m
~m~~~~mm™m~~~m~~ru~™~mm~~m~m~~~mm~~m~mm~mmmrn™~mrn~™~mm l%nn60ttnu~~~43mnu
11 1 111 11 11 1mmmmmmmmrurnmmm~~rn~mmmrnm™mrum~wmm™~™m••
~mm~rn~mm~mmmmmmm~~~mw~mmm~ru~~mmru~mmmrurnmmm~~mm~mmmmmm mmmm~m~~~mmm•rnmmm~~rnmm~mmmmmmmm•ru~~m™mm~mmm~m~m~rn~mm 426UUNttU~45~U~lltt,
1 1 1 1tt11wm~w~mruruum~mu~™m™~mru~12ffirnmrnm~~m~~mrn
rnm~~~rummmmm~~~m~™~ra~mmm~mmmmm~mmmmfflmm~~ru~ro~rum~m~mmmm
mm~~~mmm~rum~mrum~m~rum~~~m~m~~~mmmrn~m~mm™mrnm~~~~mM~~mm 1 14
8 1e 19
6
9 19 25 %
6 14 44 11
5
56 45 106 61 82 49 37 95
m
60 45 105 54 150 11 45 304 331
m
374 314 285 208 44
uo
374 604
m
1
802 368 283 ~6
m m
555 551
mrummmmmm™mmm~~mm~mmrummm~~mm~mmmm~mm~mm~~mmm~mmmmmmmm
~~mrummmmrnrn~m~mmrum~~m~mm~mrnmmmmrum~~mmm~~m~m~~mru~mrnmmm 1
1
4 10 10
3
5 11 14 15
3
8 2s
6
3
11 15 61
35 47 28 12
ss
,1 35 26
u
31 11
6 26
m m
114 1z3 188
m
126 21 361 221 366 1e, 487 224 173 479 651
m
340 339
mmmmm•rnm~m~rnmffiwmmm™mrumm~mmmmmmmmmmmmm~mmmm~m~~rnwmmmm ~~ffi~mmm~~~mfflW™fflm~ffl™mmmm~mmm~mromm~rn™m~~m~mfflm~mm~mrnmmm 1 1 4 10 to 3 5 11 14 25 3 8 25 6 1 32 25 ,1 35 41 11 22 55 97 35 26 61 31 es 6 16 119 m 114 223 188 m 126 11 362 211 366 2e, 481 224 m 479 m 1,1 340 339 mmm~m~mmmmmmm~m~m~wm~mffimmm~mmmmm~rnmmm~mmmm~~m~rnm~~m~
m~mm~™m~m~mm~~rnmmm™mmm~™mmm™mrn~mmm~m~~~rnm~m™ffi~~mrnmru~
4
2
5
5
1
2
6
8 14
2
4 14
3
1
18 14 34 20 17 16 11 32 56 20 15 35 18 51
2
1
1
3
1
1
3
4
1
1
2
1
2
1
10
3 15 105 116 132 132 112 102 75 16 216 13, 220 174 193 135 105 290 395 101 101 101
m~~~mmrnmm~mmm~mmmru™mmm~mmmmm~m~m~™mmmm~rumru~mmrnm~mmmm mmm~mm~~m™m~rufflm~mmrummmmrnmmmm™rnmm~mmmmmmm~~m~mmw~mmmm a
19 11 15
9
1 18 12 11
8 20 10 2,
2
,
u
61 11 11 "
,o
45
9 128 81 131 104
m
11 63 114 237 61 115 125
~~mmrnmmm~mmrom~~mmm~m~•mmrnru~rnmmmm~mmmlli~mmm~~~mmm~mm~m
~m~~mrnm~mrnmmmmmm~rn~m~mmm~mmffl~~~~~mrnmmmllimffl~~m~m~~mmm
4 4 1 5 4 10 , s s 4 10 11 6 5 11 , n 1 5 35 3' 45 45 38 35 Z6 5 15 48 11 61 103 48 n 103 141 36 15 15 m~mm~rnrnm~mm~~~mtt~~~~rnmmmmmmm~mrurumm~mffimm~w~~™~mmrom~m~ rnrn~mm~m~™mmffim~~~mm~mmrn~mrnrn~rnmm~~mmmmm~mmmmm~•m~mm~mm
;$'.
1 1 1 1 3 1 2 I 1 3 5 2 1 3 1 5 1 11 12 14 15 12 11 , Z 25 16 26 20 35 16 13 35 4' 12 26 26 ~mm~ru~mm~mm~m~rnnm~mm~mm~m~wrumrurummmmm~m~rnmmmmm~wm~mmm
!:!.
C
mm~~mmmm~rnmmmmm~~m~mm~ru~rummm~~mmru~~mmmm™rnmm™~rnm~mmm• 1 1 t 1 2 4 I 4 1 5 4 10 6 8 5 4 10 18 6 5 11 6 11 1 5 35 3' 45 45 38 35 Z6 5 TS 48 11 61 103 48 31 103 141 36 75 75
~mmmrn~rnmumrnmmmm~ill~mrumrummmm~rnmm~ru~~mmm~mm~mmm•mmmm~mm ~mmmmrnmm~mrn~m~mmmm~mmmmmm~mfflmm~mm~mm™mmm~m~mm~m~m1nmru 1
2
2
3
2
6
3
4
2
2
5 10
3
2
6
3
,
3 1' 22 25 26 22 20 15
1 2 2 3 2 6 3 4 2 2 5 10 3 2 6 3 9 3 1' 22 25 26 22 20 15 3 43 28 45 36 60 28 22 61 83 21 44 44 mrummmmm~~mmmmillm~mru~mm~~m~mrum•ru~m~fflwmruruffimmrn~mmm~m™~rnro
1ummmrumm~ffimmmrn~~~mmmmmm~™m~m~mmruru~~m~mm~mmrn~mrn~mm~~•m 1
1
I
3
1
2
1
1
3
5
2
t
3
1
5
1 11 12 14 14 12 11
8
ummmllimm~mmmmmmm~rn~~mmm~m~~m~~mrummmm~rum~~mrnm~~m~mmmm™ I
1
1
t
3
1
2
1
3
6
7
8
8
l
6
5
I 14
,
15 12 20
,
l
20 28
1 I I 1 3 1 t 1 2 6 7 8 8 1 6 5 I 14 , 15 12 20 , 1 20 28 1 15 15 urnnmmnm~~rn~mrnm"~mm~m~mmm~~~mmmmm~~m~rn~~m~m~rnmm~m~mmffl
mm~™mmrnmmm™~~mmrn~~mrn~rum~mm~mm~mru~~rnmmrurnm™rnm~mmmrnmm~m 1
1
1
3
3
4
4
3
3
2
8
5
8
6 11
5
4 11 16
4
8
12
!3
14
15
'
uMqmm~nmmmumm~48 e~rnm~™~mrnmm~~rnmmrnmmrnrnrumffimmm~~~mm~~~m~ ~~m•mmmmm~mw~mmrnm~*mmmruwrnmmmffim~rnmrnm~m~~rumm~™~m™~•mmrn 1 1 I 3 3 4 4 . 3 3 2 8 5 8 6 11 5 4 11 16 4 8 8 8~illffi™™~~mmm~m~mmm~ffimmm~~mmm~rnm~mmm™mm
uMq™m~nmmm~mwa~ mmm~mm•~m~mmm~m™mmmmmrn~m~mmm~m~m~mmmmmmm~™rn~mru~mmm~M 1222221
11
1 15 15
~rn%mm~mrummmm~m~nmru~m~mmm~rnm~~mmm~mm~m~mmmm~mrnmmrn~mmw Mmfflmmrn~mffl~mmmfflm~mm~~rumruffimmmmffl~~m™~mmm~™mmmmmrnmrnmmm~•
16
17
42436326,255
, ~ u 4 ' N t t ~ u a r n u ~ m 4 ' ~ 4mmfflfflm™m•mm™rnmm~m~mfflrnrn~~-~mru™mmru~mfflmm ~~mrum~~m~rnmm~mm~rnm~m~mm~rnm~m~mmm~~~m~~™m~~~~mrnmammmrn
18
11111 212231135!23 4 15 , 21 25 8 13 31 4' 12 14 32 11 21 16 a toe ,z 242 m 1" rn 10a m 441 1n 121 306 154 488 35 111 MS 14' 301 423 156 m 963 24' 445 608 m m 531 113 347 m 268 m m 043 rnrnmmm~m~rnmm~~mrn~mmmmmm~mrnrnm~~~w~fflmmmmmmrnmm~~~mrn~~~mm
1,
1
2
1
3
6
5
2
6
,
5
4
I
,
8 2' 12 24 22 11 22 55 15 11 41 15 80
3 36 120 160 16' 231 151 1'6 1'3 5, 351 233 435 318 515 21, 200 611 ,01 22, 505 5,1
m~~™m~mmmmmmrum~~mm~ffim~m~™™~mmmmmm~™~~~~mfflm~™~m~mmmmw 6~U4'Ntt~60nrn~~w~~
1222221
20
424363269255
4mm~mmmm~mm~mm~~rn~m~~mmmmmmmm~~~~mm~m
rnm~™~mm~™fflmmmfflfflrnm~~rn~~~~mmffl™~mmm~~~m~ru~mmrn~~™™mrufflm~rn
-< ~
2 25 16 26 21 35 16 13 35 4' 12 26 26
mmmmmmm~~rn~mm~m™mm™~mrnmmm~m~rnfflmmm~~m~~mm~rnmmllimm~mmma
~. --, rl-
10
3 43 28 45 36 60 28 22 61 83 21 44 44
mm~~mmffl~~m~™mmmtt~~~rnmm~~~mmm•~™m~fflmmmm~~mmmwmm~mrnmmm fflmm™~mm~mmmm~~m~~~mrnmmmmru~mru~~~m~wmmm~™mm™~~mmwmmmmm
t
'"O
21
2:: (1)
V,
Table 3.
(continued)
RL6EBRA ES, REPRESENTATIONS OF CLASS 0
0 l 01 1 2 2 401 1 Q 0001000000000000101001000020100000100000000010010000 00011000000000000010000010000000000000100000000110100010001000100000100010000000012001000020000000000000 0000100100000000000100000000000010000000000100001001 0000002010010000110000001000000001010000000100000010 0000100011100210001010000002010001000210110010101000 4 1 1 1 0 1 0 1 1 0 0 3 2 2 4 6 0 0 1 0 1 3 3 0 1 0 0 0 0 2 0 4 1 0 1 2 1 1 3 5 0 0 2 0 0 2 0 2 1 1 1 l
51
53 54 55
56
51 58
59
60
61
62 63 64 65
66 61 68 69
10
99 100 101 101 103
NUMBER
11111 212231135121 2 16 10 23 25 9 13 28 41 16 12 25 18 22 12 2 113 98 236 143 191 121 99 256 443 113 130 197 166 451 38 152 951 141 340 364 195 103 814 212 416 631 628 139 601 695 388 118 163 399 834 818 113 098 545 616 633 226 334 319 156 268 381 162 144 083 021 289 581 631 236 600 928 849 143 149 012 650 292 895 953 423 685 110 904 605 688 158 348 452 044 092 141 011 645 012 840 481 194 996 009 644 406 604
1/
11111 111231135121 2 16 10 23 25 8 13 28 42 15 12 26 19 22 12 2 113 91 231 142 191 123 100 254 444 112 129 298 164 455 38 155 956 142 331 310 189 108 882 114 414 634 635 138 591 698 380 119 176 396 841 896 811 110 543 192 741 939 413 165 316 848 515 365 118 500 315 334 100 516 046 995 462 341 m 041 233 118 159 983 843 486 014 080 688 130 504 430 330 514 244 814 119 850 135 402 528 613 143 144 519 113 150 460
23
1 1 1 1 2 2 1 1 1 6 4 9 11 3 6 14 21 34 6 14 38 12 1 1 53 46 119 68 99 66 53 123 226 84 63 156 81 241 18 89 493 603 697 148 621 613 508 130 326 880 450 161 940 931 149 054 891 114 601 673 681 930 543 982 188 822 212 280 561 121 623 100 124 642 560 554 354 101 314 684 170 493 188 592 898 492 309 205 200 516 956 120 521 890 019 048 089 004 465 413 801 852 305 318 151 685 JOB 829 202 361 140 880
24
71
12 13 14 15 16 11 18 19
80 81
82 83 84 85
86 81 88 89
90
91 92
1
93 94 95
96
1 1 1
91
98
UEl6HT
2 2
1 1
1 1 4 10 11 4 6 13 19 35 5 12 31 10 5 1 55 48 116 11 98 61 50 129 225 88 66 152 86 233 20 80 498 601 109 125 631 591 474 117 338 891 432 169 910 930 165 041 854 118 518 609 203 145 152 656 109 214 115 134 688 954 918 363 650 683 919 148 260 390 928 468 118 311 583 939 221 951 138 801 160 889 216 541 192 622 668 391 328 282 042 065 143 483 218 536 976 759 912 134 199 116 216 082
15
1 1 1 2 2 4 5 1 2 6 9 16 3 6 11 5 3 25 22 51 33 48 31 26 62 112 43 32 18 41 123 9 44 252 312 364 386 318 319 265 68 108 474 114 621 050 505 410 113 511 425 815 909 658 980 004 432 201 146 812 229 634 091 008 501 803 695 391 109 611 446 266 816 340 119 146 453 660 112 356 032 938 356 985 142 631 445 864 832 926 389 030 429 055 689 510 126 364 864 880 591 483 402 611 914
16
1 2 2 1 4 2 1 4 3 13 5 11 9 1 11 16 1 5 19 1 31 1 16 59 19 86 113 11 96 93 28 182 122 223 168 212 145 106 318 411 121 266 308 313 390 248 544 819 203 510 584 512 528 890 412 036 101 632 414 366 183 436 916 052 896 849 265 544 916 938 951 156 289 899 832 240 820 136 108 668 803 696 564 131 993 080 080 111 214 993 152 404 311 160 064
11
3
2
4
5
2
2
1
2
5
8 16
2
5 11
4
2
26 23
56
34 48
1
2
3
1
2
2
1
12 11
26
16
45 33
16
1 1 1 44 111 10 41 154 311 310 311 335 309 250 62 112 418 166 630 063 502 418 111 551 421 864 881
54 11 16
31
12 51
29 14 65 111
~mmm~mmm~mm~~~•~m~rnmmmmmmmm~154~m•~mm™m5m~mmm~~m™mm~wrn I
1
1
23 14 11
31
5 10 121 151 189 192 114 159 119
18
32 313 253 '1Q3 334 565 261 225 593 830 231 466 414
mm~mm~rn•~™m~~wmmm~mmm•mffimm&~m~~™ru™~m~am~•m~mmmm~mmm~
19
1 2 2 1 2 3 1 1 1 1 2 1 12 11 26 16 23 13 11 31 53 22 16 31 21 51 5 10 121 151 189 191 115 158 128 32 372 253 '1Q2 333 561 266 221 594 828 233 465 471 181 139 855 064 142 910 162 '1()1 112 426 181 393 110 142 118 252 143 150 158 660 184 895 949 235 886 520 914 190 619 181 641 138 631 195 882 604 261 844 934 153 981 098 360 910 588 319 811 083 135 059 038 940
30
I
I
1
2 5 3 5 2 2 1 11 5 4 1 6 12 1 4 19 31 46 45 46 38 31 8 96 66 103 88 155 10 66 161 211 61 126 121 11 154 105 384 294 238 168 315 560 344 202 356 344 351 211 56 401 548 104 416 096 660 548 525 214 314 108 958 005 159 681 368 960 905 928 180 096 921 696 358 166 555 600 368 008 036 148 128 118 368 198 504
31
1 I 2 1 2 4 1 I 2 1 2 I 11 10 26 16 22 14 12 30 54 21 16 38 21 60 5 21 116 158 181 196 171 163 135 35 311 251 406 3l3 559 268 212 594 838 230 411 481 rnrn~•~mmrnrummmm~e•mmmmmmm•m~mmmmmmmmmmmrnmma™~•mmm~~~~2H
)1
1 1 3 1 3 1 5 4 12 1 10 6 5 14 25 10 1 18 10 29 2 10 61 78 93 98 85 83 10 18 190 130 211 113 190 140 116 311 444 122 151 261 101 410 338 136 901 303 511 089 810 012 600 241 430 142 101 159 311 766 234 490 600 931 814 493 925 330 688 380 156 313 511 910 110 905 488 926 894 391 494 848 608 451 110 804 600 691 581 0% 218 350 909 814 101 145 96 108 351
1 I 1 1 1 5 2 4 4 3 5 12 3 2 9 3 16 1 28 38 42 54 39 46 44 13 92 63 111 86 140 14 56 163 241 63 138 156 19 109 594 016 012 363 912 622 824 611 159 951 119 848 800 892 166 311 439 210 903 908 246 834 841 955 460 462 810 808 022 064 156 460 461 991 518 995 836 968 019 285 312 813 115 293 114
n::, ~
l3 14
"O
rt (D
..... .i,.
1 1 3 1 3 5 4 11 1 10 6 5 15 15 10 8 11 10 18 1 10 61 18 94 91 81 81 61 11191 130 109 113 193 139 119 311 439 113 249 155 n•m•m~~mmrnmm~m•rnffl~~mmmm~mm•~mmmmmmrurummmmmffi~m•mwm•a5~• 34 45
36
34
2 1 2 1 1 1 S 1 1 4 1 1 3 13 18 20 2S 18 22 21 6 46 32 SS 43 10 31 28 64 13S 24 84 206 408 432 163 3S1 603 347 264 14 818 719 448 160 079 691 413 484 4S4 8S1 338 134 164 SOO 440 311 188 480 514 360 928 455 141 485 167 112 070 912 980 011 619 1 1 1 3 3 8 2 1 10 4 8 S 15 8 11 9 S 71 111 45 116 111 184 219 14 . 531 • 431 806 630 • 999 . 934 . 850 . 961 145 735 014 111 101 410 160 HO 519 733 410 694 708 927 009 .
17
.
11
11
40 118 109 14 64 115 120 511 16 146 544 142
1
1
2
1
31 36 J1 38
-0
39
1 2 1 2 1 1 3 5 2 1 3 2 S 2 13 11 21 12 21 18 15 4 41 33 51 44 16 35 32 81 113 34 65 66 87 11 983 014 188 411 141 194 118 165 016 355 810 631 606 714 119 114 830 905 300 154 780 968 866 218 346 173 055 356 168 602 830 696 678 043 685 834
40
-< -I p.J 2:: (l)
. 11 63
1
2
1
1
. 225
1
5
1
1
4
1
1
• 588
• 336
,
3 13
18
• S04 512 9S9
20 15 19 21
10
6 46
. 603
. 980
32 S4 43
11 36 19 82 121
32 10 17
so
34 18 119
32
80 106 370 430 135 318 620 297 1\8
H•m•m~m•m•~~~rnm~mmrnm~•mmillm*•~m~~~mmma
41
15
10
17
30
71
1 2 1 2 1 6 8 9 11 9 9 9 1 12 15 16 11 35 17 15 40 59 · 16 34 31 Z1 350 342 951 538 888 624 563 185 268 863 682 109 954 994 213 264 040 310 840 238 448 904 096 113 319 691 115 SS2 968 962 249 516 048 68\ 609 224
42
93
2 1 2 1 1 3 5 2 1 3 2 5 1 13 11 21 22 21 19 16 4 47 33 52 44 15 36 31 81 115 33 66 68 21 972 914 211 490 049 151 118 132 119 303 112 136 36S 961 616 229 149 9SO 185 S24 081 2S7 319 422 449 333 72S S86 S32 099 563 399 166 231 517 350
43
60
39
48 108 120
so
60 110 160 45
96 224 102
10 141 141 503 81 163 5SS 153
1
7 15
11
14 45
10 29
66 131 162 51 108 218 101 21
1 18 IS
40
36
12
22
21
42 10
3S
1S 63
S6
12 188 24 48 204 16
66
S6 45
18 63 21
10 11
2
1
3
• 385
1 . 099
1
18
38
S6
1S 30 12
21 20
16 11
10
31
22
21 16
16
16 16
16
11
22
9 11
9 10
9
2 21
lS
26
21
35 18 14 40
S9
16
34 37
44 45
1 1 1 1 1 2 6 8 10 10 10 8 7 2 12 16 2S 21 31 11 16 40 56 17 32 33 6 381 396 880 621 864 490 468 404 181 046 790 S11 131 SOO 318 936 204 114 392 236 188 820 416 040 194 139 323 760 416 591 176 3S6 482 002 991 611
46
1 2 3 4 4 4 3 3 10 1 11 10 18 8 7 19 27 8 16 16 1 144 152 326 1S3 340 181 188 S96 88S 463 342 631 496 002 143 311 692 580 138 500 728 889 308 904 661 761 890 314 016 435 960 S14 OS1 431 001 166
41
21
. 196
. 162 280 224
. 399
1
3
4
4
4
4
4
1 10
1 12
10
11
8
7 19 28
8 16
48
71 266 294 191 160 216 280 336
1 1 1 1 2 1 1 4 3 s 4 8 3 4 9 11 4 1 1 ~rnrnm•~mmmm•mmrnmmmm~~•m
49
80 114 88 14 180 313 146 104 322 144 S47
4 3 5 4 1 4 3 8 13 3 8 8 1 1 1 2 1 2 1 40™~m•~•rnm~m~•~m~m*rnmm•
50
. 397
I 1 1 1 1 2 1 1 2 1 4 7 1 4 6 ·™emm~mmm~~mmmm~wma•mm
51
15
1 . 544 819 103 510 584
52
1 1 1 2 1 1 1 3 7 8 4 3 s 4 8 3 3 9 13 53 186 096 S52 986 040 918 826 644 481 197 529 S48 811 141 969 606 118 147 940 893 111
\3
2 1 2 2 3 1 1 4 5 1 3 3 S6 434 610 861 198 81S 101 630 181 101 624 415 156 710 800 694 172 817 883 555 591
54
71 106
44 40 132
84 1S4 56
66
11
S1
88 14
. 149
• 150
22 17 41
48 122 88 120 16 13 113 354 169 12S 269 181 452
14 15
35
35
42 11 18
84 126 17 49
91
. 104
10 141 20
• 112
• 190 S28 313
• 390
. 148
~M
V,
17
6rnmmmmmmma~mmmmrummmmmmmmmmBrnmmmmm~mEm
12 18
8
1 1 3 1 3 4 1 4 2 6 4 6 4 2 9 16 3 9 14 • 129 910 645 344 430 117 199 206 681 004 744 930 193 091 543 443 981 S48 822 841 280
49 21
5
n~mrn~~wmwmmmm~~mmmmmm•mmmmmmm~mffi~mm™~
1
13
C
::!'.
3
. 344 019 504 191 916
29
2
32
$
2 2 S 3 4 2 2 6 11 4 3 8 5 13 1 4 29 38 46 41 43 40 33 9 96 66 106 88 150 71 61 161 227 64 130 134 56 333 174 312 410 764 973 599 915 797 985 136 317 061 317 303 944 620 105 134 591 332 331 972 087 145 675 415 9S1 116 843 6SS 026 803 SS1 321 140
56
1 1 2'I 175 119 300 339 134 193 401 610 281 211 453 410 411 153 55
81 122 32 70 78 824 307 250 890 513 21 35 8 10 31 111 424 141 694 668
w N
th
Table 3.
(continued}
ALGEBRA EB, REPRESENTATIONS OF CLASS 0
0
4
1
1
0001000000000000101001000010100000100000000010010000 00011000000000000010000010000000000000100000000110100010001000100000100010000000011001000010000000000000 0000100100000000000100000000000010000000000100001001 0000001010010000110000001000000001010000000100000010 0000!0001!300110001010000001010001000110110010101000 4 1 1 1 0 1 0 2 1 0 0 3 1 1 4 6 0 0 1 0 1 3 3 0 1 0 0 0 0 1 0 4 1 0 1 1 1 1 3 5 0 0 1 0 0 1 0 1 1 1 1 3 UEl6HT
51 53
54 55
56
51
58 5,
60
61
61 '3 64 65
66 61 68 6'
15 11 4
4
70
71 73
14 75
30
36 15 22
75 133 61
12
15
10 10 20
4 25 18 25 22
90
81 81 83
10 10 12
48 20 10
16 3()
46 109 66 184 20
10 10
51
4
4
16 77 18 1!
36 1, 48 21 11 !j() 114 10 56 85 ,3 141 32
15 14 40
4
11
93 94 '15
96
91
98
99 100 \01 !OZ 103
2
1
2
2
3
1
1
4
5
1
3
3
2
1
2
1
3
1
1
4
6
1
3
3
"~~~rn~~~m~m~mmmmm~rnmm
NUMBER
55 56
51
I 1 1 1 I I 10 18 152 250 320 330 310 326 304 101 885 618 035 960 552 196 104 144 118 161 113 144
58
• 131
1 I 1 3 1 1 81 139 275 231 510 220 506 60! 238 184 562 230 81' 212 848 53, !j()4 246 111 020 610
59
44
1 1 18 44 110 80 180 15 210 231 101 325 242 410 384 528 338 224 136 402 303 904 089
60
3() 51
1 1 1 2 1 1 10 21 160 1~ 335 307 359 282 254 77 879 682 015 ,21 625 166 118 821 563 851 588 611
61
36 45 64 64 36 24 64 110 14 1'1 106
62
12
60 00 120 96 224 81
20 21 15
!j() !I '1
~~m~mmm~mmmmmrnm~mffima~
11
38 16 66
40 16 19 31
86 BT 88 89
11 304 125 330 328 832 240 563 864 869 563 558 520
32
53
84 85
15
28
36
11 50 100
12
20
54 S4 34 45
10
6T 170 101
19 109 594
63
1 1 90 177 92 116 204 77 31' 235 480 314 556 341 251 000 313 350 84! 016
64
14 84 106
65
10
• 145
21 S4 34 11
96
45
36
. 208 351
64 I~
11
34
66 61
10
11
II
15
54 81 118 105 138 4
4
10
,8
18 14 44 30 12 26
1 88 11 353 185 408 384 618 321 334 758 066 364 611 671
68
6 120 96 135 132 312 106 19' 336 360 101 231 211
69
22
m
1S4 222 12, 102 3()0 534 114 353 401
10
24
1 134 120 153 148 268 132 136 3()4 405 154 158 152
11
11
6 40 30 55
40 51 118 114 14 114 115
11
15
~
30 56 31 61
68 11 125 95
16
23
48 30 52 21
10 10 16 16 16
S4 96
n:::r s:,.,
,....
"O (t) ....,
.j:,.
n
4
u
u
4 "
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H
u
•u n 4
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u
u u
~
u ~
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n ~ M n u M u u u u
"M "
~ ~
"
~
M Q ~
~
n u
u
4
a
M u u 4
4
4 4
u
R
hN
u
~
~
N
~mm n ~ q ~mm ~!Mia
n
M
~m R
"
R
n n n ~
Q
~
Q
~
""
M
~
"
M
~
~
u u
n
~
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~
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4
8
~
"
n n n
H
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H
~
"n
~
~
n
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n
~
U
~
M
~ ~
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ro
~
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4 U
u
~
4
£2. ~
M
Q
u
tt
u
~
11
u e 8
"
~ C
" "u 9
~ ~
~
4
s
~ ~
n ~ ~ ~
~
%
n ~ ~
100
m 1~
m
w
~ ~
Table 3.
(continued)
AL6EBRA F4, REPRESENTATIONS OF CLASS 0
w
8 H 12 UM U U
~
22
M H H 28 32 36
~
N N
~
36
~
40 42 42
24 24 96
24 144 96 24 192 24 288 144 288 96 144 192 192 24 M 516 144 288 288 288 192 144 192 516
11 23
33 39 43 45
31
41
49 53 55 59
61
61 63 65 65
61
0
~
44
~
48 46
~
~
64
~
M
~
M
~
~
~
62 H
~
M 288 144 96 516 192 144 24 516 96 516 288 192 144 516 192 288 516
11 13 15 15 11 11 19 81 81
83 83 85 85
81
81 89 89 91
91
91
93 93 95
95
12 62
~
H H
~
10 00
SY.
24 516 144 288 516 188 0 .S.
91 91 91 99
99 101 101 103 LEVEL
1 1 111112 31243 I 1 1 4 2 8 10 19 19 11 34 2' 16 12 106 16 101 119 160 205 160 212 319 81 420 184 226 181 340 412 100 118 629 311 952 341 002 801 042 014 192 342 921 484 488 313 508 1 26 52 213 324 053 214 053 096 652 424 829 218 448 901 149 112 302 316 496 016 406 119 056 151 056 992 848 081 147 156 146 644 119 116 116 208 356 104 952 522 456 311 899 944 556 056 312 406 563 088 596 0 In'"
1 0
1 0
)
)
1
0 2
1)
U[ I6HT
9 10 11 12 13
4
14 15 16 11 18 19 10 21 22
23 24 25 26 21 28
29 30 31
32
33
34 35 36 31
38
39 40 41 42
43
44 45 46 41
48
49 SO 51 NUn8£R
11 21 26 21
64
10 3 6 11 8 4 2 4 3 4 3 5 3 2 8 1 3 1 1 1 1 1 36 96 125 114 116 141 285 228 126 8 8 • ~ r u m ~ r n l l i ~ ~ ~ m m ~ r n m m m m m m m m m m m ~ ~ ™ r n m m m ™
14 13
40
24
1
10 4
4
"
17 126 111 106 201 141 84
24 13 40 56 80 84 66 138 118 62 14
I
I
2
3
1
3
2
3
2
4
2
2
6
8
3
5
9
1
~m™ru~~~rum~m~mmmmmmmm™m~m~mm~IBm™rnrnm 41 368 239 330 338 496 m
1 2 4 1 5 2 I 2 I 2 I 3 2 I 5 I I 465 584 964 1,e 014 416 552 ,04 784 954 211 488 324 768 995 166 958 436 ooe 868 289 610 016 m 200 412 a,o
21 31 51
~
50
95 12 41
5 2 3 6 4 I 1 1 2 1 2 1 2 1 1 4 21 261 181 241 258 351 444 329 438 155 156 199 338 404 461 510 120 149 940 050 118 593 168 590 114 60(1 531 153 481 718 036 454 042 116
15 23 33
3'
29 64 58 31
4 1 2 4 J 1 1 1 ! 1 2 I 1 3 1 M 184 122 113 181 261 324 260 318 545 116 613 242 331 116 411 515 134 504 813 104 120 1Z4 221 161 212 204 382 400 554 694 146 811 813 1
10 12 14 23 13 42 34 19 14 i1e 92 128 139 184 138 119 240 421 13 16
12 21 18 14 13 88 59 81 96 140 169 141 116 301
8' 456 200 211 849 343 426
1
1
1
I
2
18 14
4
4
4
40 26 42 52 6
s,
68
I
1
3
l
1 1 2 2 1 I 3 2 1 63 354 140 194 637 28' 339 85 890 489 034 131 041 141 348 181 111 133 253 886 IOI 110 128 519 1
14
4
3
~mmmmmmw~mmmrum~•m
83 18 88 164 33 1'5 84 114 351 117 192
I
I
I
I
~m~~mrnm~mmmmmmmm~
1 I 2 ! 1 I 45 63 11 93 124 93 130 132 51 251 113 134 480 198 249 56 678 382 195 580 806 603 012 623 614 633 199 301 841 441 499 998
14 23 28 31
1 I I 45 63 45 11 110 32 133 61 13 164 111 146 34 384 124 456 346 484 363 638 392 380 992 132 421 511 905 584 169
16 20 30 31
44 60
48 64 115
I I I 1 H 136 66 75 262 1!6 138 35 386 221 412 335 410 3~1 654 310 398 000 121 423 531 913 561 172
10
n:::, I")
"O
16 12 18 22
I I I 34 41 31 48 80 19 103 41 61 191 '1 113 n m m m ™ I D m ~ r n m m w m ~ m ™ r n
II
.....
10
2Q
19 22 32 63
14 10 34 40 139 63 16 10 214 126 266 193 113 212 384 112 239 591 15 865 318 566 968 191
12
.JO,.
IS
19 18 22 39
13 16
4 4
10
4
54 60 19 158 91 198 136 211 150 284 169 118 480 62 660 211 429 756 642
13
II
12 12 33
36 23 21 69
39 37 14 112 61 160 100 144 124 230 119 160 352 42 513 215 341 572 488
14
13
18 28
32 16 19 12
30
10 114 70 141 111 159 120 216 140 138 346 SI 510 192 340 600 486
IS
53 20 34
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Table 3.
(continued)
ALGEBRA F4, REPRESENTATIONS OF CLASS 0
10M12u10l6MNl6"~~™~"uu~•""""88"rn"n~~~rn~•""m™mrnmmmmmmmmmmmrns,. 1 1 576 144 288 1-14 516 192 1'2 576 288 192 192 144 24 152 1'2 288 516 576 24 288 516 144 576 288 516 288 516 576 144 96 516 144 576 576 1,2 576 288 516 516 1'2 144 96 288 1-14 152 1'2 1,2 576 516 576 24 192 0 .s.
103 103 105 105 105 101 101 101 101 109 109 109 111 111 111 113 113 113 113 115 115 115 111 111 111119 11' 119 119 121 121 121 121 123 123 123 123 125 125 12s 115 121 121 121 121111 129 129 129 12, 119 131 3
1
4
1
1
3
5
8
4
1
6
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16
4 10 10 15
1
9 23
8 16 15 1'
l
28 26
1 11 38 10 34 31 28 44 18 46 28 21 20 13 33
1 91 15 26 18 6' 65
um
4 16
mrnwm1umm~mm~~~mru~~m™mmm~mmw~mmm~~~•~m™w~m~m~mmmm~m~mrn
952 212 656 416 106 192 304 094 953 163 856 '16 912 216 088 412 592 926 120 388 488 116 831 6!16 624 49' 544 956 !