Polynomial Representations of GLn : With an Appendix on Schensted Correspondence and Littelmann Paths

by Erdmann, K. H.; Green, J. A.; Schocker, M.

• ISBN13: 9783540469445
• ISBN10: 3540469443
• Edition: 2nd
• Format: Paperback
• Copyright: 2007-02-03
• Publisher: Springer Verlag

Summary

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 symmetric groups, has been the basis of much research in representation theory.The second half is an Appendix, and can be read independently of the first. It is an account of the Littelmann path model for the case gln. In this case, Littelmann's 'paths' become 'words', and so the Appendix works with the combinatorics on words. This leads to the representation theory of the 'Littelmann algebra', which is a close analogue of the Schur algebra. The treatment is self-contained; in particular complete proofs are given of classical theorems of Schensted and Knuth.

Table of Contents

Polynomial representations of GLn
Introduction	p. 1
Polynomial Representations of GLn (K): The Schur algebra	p. 11
Notation, etc	p. 11
The categories MK(n), MK (n, r)	p. 12
The Schur algebra SK(n, r)	p. 13
The map e: K¿ → SK (n, r)	p. 14
Modular theory	p. 16
The module E2+⊗r	p. 17
Contravariant duality	p. 19
AK(n, r) as KT-bimodule	p. 21
Weights and Characters	p. 23
Weights	p. 23
Weights spaces	p. 23
Some properties of weight spaces	p. 24
Characters	p. 26
Irreducible modules in MK(n, r)	p. 28
The modules D¿, K	p. 33
Preamble	p. 33
¿-tableaux	p. 33
Bideterminants	p. 34
Definition of D¿, K	p. 35
The basis theorem for D¿, K	p. 36
The Carter-Lusztig lemma	p. 37
Some consequences of the basis theorem	p. 39
James's construction of D¿, K	p. 40
The Carter-Luszting modules V¿, K	p. 43
Definition of V¿, K	p. 43
V¿, K is Carter-Luszting's "Weyl module"	p. 43
The Carter-Lusztig basis for V¿, K	p. 45
Some consequences of the basis theorem	p. 47
Contravariant forms on V¿, K	p. 48
Z-forms of V¿, K	p. 50
Representation theory of the symmetric group	p. 53
The functor f: MK (n, r) → mod KG(r) (r ≤ n)	p. 53
General theory of the functor f: mod S mod eSe	p. 55
Application I. Specht modules and their duals	p. 57
Application II. Irreducible KG(r)-modules, char K = p	p. 60
Application III. The functor f: MK (N, r) → MK (n, r) (N ≥ n)	p. 65
Application IV. Some theorems on decomposition numbers	p. 67
Appendix: Schensted correspondence and Littelmann paths
Introduction	p. 73
Preamble	p. 73
The Robinson-Schensted algorithm	p. 74
The operators &ecedil;c, &fbar;c	p. 75
What is to be done	p. 78
The Schensted Process	p. 81
Notations for tableaux	p. 81
The map Sch: I(n, r) → T(n, r)	p. 81
Inserting a letter into a tableau	p. 82
Examples of the Schensted process	p. 85
Proof that (¿, U, V) ← x1 belongs to T(n, r)	p. 88
The inverse Schensted process	p. 89
The ladder	p. 92
Schensted and Littelmann operators	p. 95
Preamble	p. 95
Unwinding a tableau	p. 96
Knuth's theorem	p. 103
The "if" part of Knuth's theorem	p. 107
Littelmann operators on tableaux	p. 114
The proof of Proposition B	p. 116
Theorem A and some of its consequences	p. 121
Ingredients for the proof of Theorem A	p. 121
Proof of Theorem A	p. 124
Properties of the operator C	p. 127
The Littelmann algebra L(n, r)	p. 129
The modules MQ	p. 131
The ¿-rectangle	p. 134
Canonical maps	p. 135
The algebra structure of L(n, r)	p. 137
The character of M¿	p. 139
The Littlewood-Richardson Rule	p. 140
Lascoux, Leclerc and Thibon	p. 143
Tables	p. 147
Schensted's decomposition of I(3,3)	p. 147
The Littelmann graph I(3,3)	p. 148
Index of symbols	p. 151
References	p. 155
Index	p. 159
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