Polynomial Representations of GLn : Lecture Notes in Mathematics - K. Erdmann

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Polynomial Representations of GLn

By: K. Erdmann, James A. Green

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Paperback | 1 February 2007 | Edition Number 2

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180 Pages

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Revised

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22.86 x 15.24 x 1.27

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This second edition of "Polynomial representations of GL (K)" consists of n two parts. The ?rst part is a corrected version of the original text, formatted A in LT X, and retaining the original numbering of sections, equations, etc. E The second is an Appendix, which is largely independent of the ?rst part, but whichleadstoanalgebraL(n, r), de?nedbyP.Littelmann, whichisanalogous to the Schur algebra S(n, r). It is hoped that, in the future, there will be a structure theory of L(n, r) rather like that which underlies the construction of Kac-Moody Lie algebras. We use two operators which act on "words". The ?rst of these is due to C. Schensted (1961). The second is due to Littelmann, and goes back to a1938paperbyG.deB.Robinsonontherepresentationsofa?nitesymmetric group.Littelmann'soperatorsformthebasisofhiselegantandpowerful"path model" of the representation theory of classical groups. In our Appendix we use Littelmann's theory only in its simplest case, i.e. for GL . n Essential to my plan was to establish two basic facts connecting the op- ations of Schensted and Littelmann. To these "facts", or rather conjectures, I gave the names Theorem A and Proposition B. Many examples suggested that these conjectures are true, and not particularly deep. But I could not prove either of them.

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From the reviews: LNM 830 "is now regarded as the standard text on the finite-dimensional polynomial representations of the general linear group GL_n(K)."

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Polynomial representations of GL_n
Introduction	p. 1
Polynomial Representations of GL_n (K): The Schur algebra	p. 11
Notation, etc	p. 11
The categories M_K(n), M_K (n, r)	p. 12
The Schur algebra S_K(n, r)	p. 13
The map e: K → S_K (n, r)	p. 14
Modular theory	p. 16
The module E^2+⊗r	p. 17
Contravariant duality	p. 19
A_K(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 M_K(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: M_K (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: M_K (N, r) → M_K (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 M_Q	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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Polynomial Representations of GLn

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