Representations of Finite Classical Groups A Hopf Algebra Approach
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869 Andrey V. Zelevinsky
Representations of Finite Classical Groups A Hopf Algebra Approach
Springer-Verlag Berlin Heidelberg New York 1981
Author
Andrey V. Zelevinsky Institute of Physics of the Earth Department of Applied Mathematics B. Grouzinskaya 10,123810 Moscow, USSR
AMS Subject Classifications (1980): 16A24, 20C30, 20G05, 20G40
ISBN 3-540-10824-6 Springer-Verlag Berlin Heidelberg New York ISBN 0-387-10824-6 Springer-Verlag New York Heidelberg Berlin This work is subject to copyright. All rights are reserved, whether the whole or part of the material is concerned, specifically those of translation, reprinting, re-use of illustrations, broadcasting, reproduction by photocopying machine or similar means, and storage in data banks. Under § 54 of the German Copyright Law where copies are made for other than private use, a fee is payable to "Verwertungsgesellschaft Wort", Munich.
© by Springer-Verlag Berlin Heidelberg 1981 Printed in Germany Printing and binding: Beltz Offsetdruck, Hemsbach/Bergstr. 2141/3140-543210
Contents ••••••••••••
1
Chapter I. Structural theory of PSH-algebras •••••••••••
12
Introduction •••••••.•
..............
12
The decomposition theorem •••••••.• ' •.••••••
21
§ 1. Definitions and first results
s 2. s 3.
Universal PSH-algebra: the uniqueness theorem and the Hopf algehra structure
s 4.
.............
27
Universal PSH-algebra: irreducible elements••
49
Chapter II. First applications •••••••••.•••••••••••••••
71
s 5.
Symmetric polynomials •••••••••••••••••.•••••
71
......... Representations of wreath products .........
§ 6. Representations of symmetric groups §
7.
86
93
Chapter III. Representations of general linear and affine groups over finite fields ••••••••••••••••••• 107 § 8. Functors §
i u,Q 0 and
r u,Q a
•••••••••• ••• ••••
107
9. The classification of irreducible representations of
GL(n,Fq) •••••.•••••••.•.•.•••.•••• 110
§10. The P.Rall algebra •••••••••.•••.•.•.•.•••••• 115 §11. The
character values of
GL(n,Fq)
at unipotent
elements •••••••••••.•••••••••••.••.•.•••••• 128 §12. Degenerate Gelfand-Graev modules •••••••••••• 138
IV
§ 13. Representations of general affine groups
and the branching rule •••••••••••••••••
143
Appendix 1.
Elements of the Hopf algebra theory •••.
149
Appendix 2.
A combinatorial proposition ••••••••••••
155
Appendix 3.
The composition of functors
i.
167
•.•....
177
Index of Notation ••••••••••••.•••.••••.•••••••••••••
180
Index ••••.•....•.•.•••.•........•...••..••....•••••.
182
References •••••.•.•.••
rand
Introduction In this work we develop a new unified approach to the representation theory of symmetric groups and general linear groups over finite fields. It gives an explanation of the well known non-formal statement that the symmetric group is "the general linear group over the (non-existent) one element field". This approach is based on the structural theory of a certain class of Hopf algebras. The original plan of this work was to apply the technique developed by J.N.Bernstein a
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