By James A. Green, Manfred Schocker, Karin Erdmann (auth.)

The first half this publication includes the textual content of the 1st version of LNM quantity 830, Polynomial Representations of GL_{n}. This vintage account of matrix representations, the Schur algebra, the modular representations of GL_{n}, and connections with symmetric teams, has been the foundation of a lot study in illustration theory.

The moment part is an Appendix, and will be learn independently of the 1st. it really is an account of the Littelmann direction version for the case gl_{n}. thus, Littelmann's 'paths' turn into 'words', and so the Appendix works with the combinatorics on phrases. This results in the repesentation idea of the 'Littelmann algebra', that's a detailed analogue of the Schur algebra. The therapy is self- contained; particularly whole proofs are given of classical theorems of Schensted and Knuth.

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N}, let cµν ∈ K Γ be the function which associates to each g ∈ Γ its (µ, ν)-coeﬃcient gµν . Denote by A or AK (n) the K-subalgebra of K Γ generated by the functions cµν (µ, ν ∈ n); the elements of A are, by deﬁnition, the polynomial functions on Γ. Since K is inﬁnite, the cµν are algebraically independent over K, so that A can be regarded as the algebra of all polynomials over K in n2 “indeterminates” cµν (µ, ν ∈ n). For each r ≥ 0 we denote by AK (n, r) the subspace of A consisting of the elements expressible as polynomials which are homogeneous of degree r 2 as K-space; in parin the cµν .

Young [58, 1902]. The observation that the Dλ,K can be constructed over an arbitrary ﬁeld—or equivalently that the (Tl : Ti ) generate a Z-form Dλ,Z in Dλ,Q —was made by G. Higman [23, 1965]. The Vλ,K (and the Z-form Vλ,Z ) were constructed, independently of all this, by R. Carter and G. Lusztig [6, 1974]. They called these “Weyl modules”, and their construction was based on methods used in the theory of semisimple algebraic groups. Towber [52] showed that Dλ,K and Vλ,K are dual to each other—his framework is “functorial” and more general than ours.

Deruyts [13], in 1892. Although Schur refers to two later papers of Deruyts, there is no sign in [47] that he appreciated that Deruyts had really given a complete set of irreducible modules in MC (n, r). The discovery of the basis of the “standard” (Tl : Ti ), seems to go back to A. Young [58, 1902]. The observation that the Dλ,K can be constructed over an arbitrary ﬁeld—or equivalently that the (Tl : Ti ) generate a Z-form Dλ,Z in Dλ,Q —was made by G. Higman [23, 1965]. The Vλ,K (and the Z-form Vλ,Z ) were constructed, independently of all this, by R.