Table Of ContentPU 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
I
MULTIPLICITIES FOR
REPRESENTATIONS
OF SIMPLE
LIE ALGEBRAS
M. R. Bremner
R.V Moody
J. Patera
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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MONOGRAPHS AND TEXTBOOKS IN
PURE AND APPLIED MATHEMATICS
1. K. Yano, Integral Formulas in Riemannian Geometry (1970) (out of print)
2. S. Kobayashi, Hyperbolic Manifolds and Holomorphic Mappings (1970)
(out of print)
3. V. S. Vladimirov, Equations of Mathematical Physics (A. Jeffrey, editor;
A. Littlewood, translator) (1970) (out of print)
4. B. N. Pshenichnyi, Necessary Conditions for an Extremum (L. Neustadt,
translation editor; K. Makowski, translator) (197 I)
5. L. Narici, E. Beckenstein, and G. Bachman, Functional Analysis and
Valuation Theory ( 197 l)
6. D.S. Passman, Infinite Group Rings (1971)
7. L. Dornhoff, Group Representation Theory (in two parts). Part A:
Ordinary Representation Theory. Part B: Modular Representation Theory
(1971, 1972)
8. W. Boothby and G. L. Weiss (eds.}, Symmetric Spaces: Short Courses
Presented at Washington University (1972)
9. Y. Matsushima, Differentiable Manifolds (E.T. Kobayashi, translator)
(1972) .
IO. L. E. Ward, Jr., Topology: An Outline for a First Course (1972) (out of
print}
11. A. Babakhanian, Cohomological Methods in Group Theory (1972)
12. R. Gilmer, Multiplicative Ideal Theory (1972)
13. J. Yeh, Stochastic Processes and the Wiener Integral (1973) (out of print)
14. J. Barros-Neto, Introduction to the Theory of Distributions (1973)
(out of print}
15. R. Larsen, Functional Analysis: An Introduction (1973) (out of print)
16. K. Yano atid S. Ishihara, Tangent and Cotangent Bundles: Differential
Geometry (1973) (out of print}
1 7. C. Procesi, Rings with Polynomial Identities ( 197 3)
18. R. Hermann, Geometry, Physics, and Systems (1973)
19. N. R. Wallach, Harmonic Analysis on Homogeneous Spaces (1973)
(out of print)
20. J. Dieudonne, Introduction to the Theory of Formal Groups (1973)
21. l Vaisman, Cohomology and Differential Forms (1973)
22. B. -Y. Chen, Geometry of Submanifolds (1973)
23. M. Marcus, Finite Dimensional Multilinear Algebra (in two parts) (1973,
1975)
24. R. Larsen, Banach Algebras: An Introduction (1973)
25. R. 0. Kujala and A. L. Vitter (eds.}, Value Distribution Theory: Part A;
Part B: Deficit and Bezout Estimates by Wilhelm Stoll (1973)
26. K. B. Stolarsky, Algebraic Numbers and Diophantine Approximation (1974)
27. A. R. Magid, The Separable Galois Theory of Commutative Rings (1974)
28. B. R. McDonald, Finite Rings with Identity (1974)
29. J. Satake, Linear Algebra (S. Koh, T. A. Akiba, and S. Ihara, translators)
(1975)
30. J. S. Golan, Localization of Noncommutative Rings (1975)
31. G. Klambauer, Mathematical Analysis (1975)
32. M. K. Agoston, Algebraic Topology: A First Course (1976)
33. K. R. Goodearl, Ring Theory: Nonsingular Rings and Modules (1976)
34. L. E. Mansfield, Linear Algebra with Geometric Applications: Selected
Topics (1976)
35. N. J. Pullman, Matrix Theory and Its Applications (1976)
36. B. R. McDonald, Geometric Algebra Over Local Rings (1976)
37. C. W. Groetsch, Generalized Inverses of Linear Operators: Representation
and Approximation (1977)
38. J. E. Kuczkowski and J. L. Gersting, Abstract Algebra: A First Look (1977)
39. C. 0. Christenson and W. L. Voxman, Aspects of Topology (1977)
40. M. Nagata, Field Theory (1977)
41. R. L. Long, Algebraic Number Theory (1977)
42. W. F. Pfeffer, Integrals and Measures (1977)
43. R. L. Wheeden and A. Zygmund, Measure and Integral: An Introduction to
Real Analysis (1977)
44. J. H. Curtiss, Introduction to Functions of a Complex Variable (1978)
45. K. Hrbacek and T. Jech, Introduction to Set Theory (1978)
46. W. S. Massey, Homology and Cohomology Theory (1978)
47. M. Marcus, Introduction to Modern Algebra (1978)
48. E. C. Young, Vector and Tensor Analysis (1978)
49. S. B. Nadler, Jr., Hyperspaces of Sets (1978)
50. S. K. Segal, Topics in Group Rings (1978)
51. A. C. M. van Rooij, Non-Archimedean Functional Analysis ( 1978)
54. L. Corwin and R. Szczarba, Calculus in Vector Spaces (1979)
53. C. Sadosky, Interpolation of Operators and Singular Integrals: An
Introduction to Harmonic Analysis (1979)
54. J. Cronin, Differential Equations: Introduction and Quantitative Theory
(1980)
55. C. W. Groetsch, Elements of Applicable Functional Analysis (1980)
56. I. Vaisman, Foundations of Three-Dimensional Euclidean Geometry (I 980)
57. H. I. Freedman, Deterministic Mathematical Models in Population Ecology
(1980)
58. S. B. Chae, Lebesgue Integration (1980)
59. C. S. Rees, S. M. Shah, and C. V. Stanojevic, Theory and Applications of
Fourier Analysis (19 81 )
60. L. Nachbin, Introduction to Functional Analysis: Banach Spaces and
Differential Calculus (R. M. Aron, translator) (1981)
61. G. Orzech and M. Orzech, Plane Algebraic Curves: An Introduction
Via Valuations (1981)
62. R. Johnsonbaugh and W. E. Pfaffenberger, Foundations of Mathematical
Analysis (1981)
63. W. L. Voxman and R.H. Goetschel, Advanced Calculus: An Introduction
to Modern Analysis (1981)
64. L. J. Corwin and R. H. Szcarba, Multivariable Calculus (1982)
65. V. I. Jstratescu, Introduction to Linear Operator Theory (198 I)
66. R. D. Jarvinen, Finite and Infinite Dimensional Linear Spaces: A
Comparative Study in Algebraic and Analytic Settings ( 1981)
67. J. K. Beem and P. E. Ehrlich, Global Lorentzian Geometry ( 1981)
68. D. L. Armacost, The Structure of Locally Compact Abelian Groups (1981)
69. J. W. Brewer and M. K. Smith, eds., Emmy Noether: A Tribute to Her Life
and Work (1981)
70. K. H. Kim, Boolean Matrix Theory and Applications (1982)
71. T. W. Wieting, The Mathematical Theory of Chromatic Plane Ornaments
(1982)
72. D. B. Gauld, Differential Topology: An Introduction (1982)
73. R. L. Faber, Foundations of Euclidean and Non-Euclidean Geometry (1983)
74. M. Carmeli, Statistical Theory and Random Matrices (1983)
75. J. H. Carruth, J. A. Hildebrant, and R. J. Koch, The Theory of
Topological Semigroups ( 1983)
76. R. L. Faber, Differential Geometry and Relativity Theory: An
Introduction (1983)
77. S. Barnett, Polynomials and Linear Control Systems (1983)
78. G. Karpilovsky, Commutative Group Algebras (1983)
79. F. Van Oystaeyen and A. Verschoren, Relative Invariants of Rings: The
Commutative Theory (1983)
80. I. Vaisman, A First Course in Differential Geometry (1984)
81. G. W. Swan, Applications of Optimal Control Theory in Biomedicine (1984)
82. T. Petrie and J. D. Randall, Transformation Groups on Manifolds (1984)
83. K. Goebel and S. Reich, Uniform Convexity, Hyperbolic Geometry, and
Nonexpansive Mappings (1984)
84. T. Albu and C. Niistiisescu, Relative Finiteness in Module Theory (1984)
85. K. Hrbacek and T. Jech, Introduction to Set Theory, Second Edition,
Revised and Expanded ( 1984)
86. F. Van Oystaeyen and A. Verschoren, Relative Invariants of Rings: The
Noncommutative Theory (1984)
87. B. R. McDonald, Linear Algebra Over Commutative Rings (1984)
88. M. Namba, Geometry of Projective Algebraic Curves (1984)
89. G. F. Webb, Theory of Nonlinear Age-Dependent Population
Dynamics (I 9 8 5)
90. M. R. Bremner, R. V. Moody, and J. Patera, Tables of Dominant Weight
Multiplicities for Representations of Simple Lie Algebras (1985)
91. A. E. Fekete, Real Linear Algebra (1985)
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 repre
sentations of simple Lie algebras.
(Monographs and textbooks in pure and applied mathe
matics ; v. 90)
Bibliography: p.
1. Lie algebras-Tables. 2. Representations of
algebras-Tables. 3. Lie groups-Tables. 4. Repre
sentations 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. ALL RIGHTS RESERVED
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PRINTED IN THE UNITED STATES OF AMERICA
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 particu
lar 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 par
ticular 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 repres
entations of £ in this volume took about 18 minutes of computing time on a Cyber
8
835) and there is no difficulty either in extending the rank or the number of repres
entations 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 litera
ture. 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 bio
graphical 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 Intro
duction 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
