Polynomial Representations of GLn
The first half of this book contains the text of the first edition of LNM volume 830, Polynomial Representations of GLn. This classic account of matrix representations, the Schur algebra, the modular representations of GLn, and connections with symme
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830 J. A. Green
Polynomial
Representations of GL n
Springer-Verlag Berlin Heidelberg New York 1980
Author James A. Green Mathematics Institute University of Warwick Coventry CV4 7AL England
AMS SubjeCt Classifications (1980): 20 C30, 20 G05
ISBN 3-540-10258-2 Springer-Verlag Berlin Heidelberg NewYork ISBN 0-387-10258-2 Springer-Verlag NewYork 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 the publisher, the amount of the fee to be determined by agreement with the publisher. © by Springer-Vertag Berlin Heidelberg 1980 Printed in Germany Printing and binding: Beltz Offsetdruck, Hemsbach/Bergstr. 2141/3140-543210
These lectures were given at Yale University semester
1980, while the author was receiving
and hospitality Foundation
of Yale University
versity of Essen,
Germany.
the kind support
and the National
(NSF contract MCS 79-04473).
given at the University of Warwick,
during the spring Science
Earlier versions were
Engl.and and at the uni-
POLYNOMIAL REPRESENTATIONS OF
GL
n
Table of Contents
Chapter i.
Introduction
Chapter 2.
Polynomial representations of GL (K) : The Schur algebra
18
2.1
Notation, etc.
18
2.2
The categories MK(n),MK(n,r)
2.3
The Schur algebra SK(n,r)
19 21 23
2.4
The map
2.5
Modular theory
25
2.6
The module E mr
2.7
Contravariant duality
31
2.8
AK(n,r) as KF-bimodule
34
Weights and characters
36
3.1
Weights
36
3.2
Weight spaces
37
Chapter 3.
e:KF+SK(n,r)
27
3.3
Some properties of weight spaces
38
3.4
Characters
40
3.5
Irreducible modules in ~_(n,r) K-
44
The modules DX, K
5O
4.1
Preamble
50
4.2
~-tableaux
50
4.3
Bideterminants
4.4
Definition of
Chapter 4.
DX, K
51 53
Vt
4.5
The basis theorem for DE, K
55
4.6
The Carter-Lusztig lemma
57
4.7
Some consequences of the basis theorem
59
4.8
James's construction of Di, K
61
The Carter-Lusztig modules VI, K
65
5.1
Definition of V%, K
65
5.2
VI, K is Carter-Lusztig's "Weyl module"
65
5.3
The Carter-Lusztig basis for V~, K
68
Contravariant forms on VI, K
73
Representation theory of the symmetric group
80
6.1
The functor
80
6.2
General theory of the functor f:mod S ÷ mod eSe
83
6.3
Application I. Specht modules and their duals
88
6.4
Application II. Irreducible KG(r)-medules, char K = p
93
6.5
Application III. The functor MK(N,r) + MK(n,r) (N L n )
102
6.6
Application IV. Some theorems on decomposition numbers
107
Chapter 5.
5.4 5.5 5.6
Chapter 6.
Some consequences of the basis theorem
Z-forms of V~,Q
f:MK(n,r) ÷ mod KG(r)(r ~ n)
70
77
Bibliography
113
Index
117
§I.
Introduction Issai Schur determined the polynomial replesentations of the complex
general
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