Determining Irreducible GL(n,K)-Modules

In this paper we consider several different methods to produce spanning sets for irreducible polynomial representations of GL(n,K) for an infinite field K, and we show how these spanning sets are related. The irreducible polynomial representations of GL(n,K) can be afforded by submodules L(λ) of Schur modules ∇(λ), indexed by partitions λ of positive integers r. Each ∇(λ) is a GL(n,K)-submodule of the polynomials A(n, r) of degree r in the n coordinate functions xij on GL(n,K), where GL(n,K) acts on A(n, r) by right translation. The module ∇(λ) has a K-basis consisting of bideterminants corresponding to semistandard λ-tableaux. The module L(λ) is generated as a GL(n,K)-module by a highest-weight vector Tλ, which is a product of determinants of principal minors of the matrix X = (xij)1≤i,j≤n. If K has characteristic 0, it is well known that the modules ∇(λ) are irreducible, that is, ∇(λ) = L(λ). If the characteristic of K is p > 0, then in general the dimension of L(λ) and the dimensions of its weight spaces are not known. We give several methods for finding K-spanning sets for L(λ), all of which are adapted for the weight-space decomposition of L(λ). Our first spanning set B comes from evaluating bideterminants at XA, where A is an element of GL(n,K), using the Binet-Cauchy formula. This is then compared to a spanning set of L(λ) produced by a method due to Pittaluga and Strickland in [PS], which is given as follows. For a partition λ whose first part λ1 = s, let λ̃ be the partition which complements λ inside the rectangular Young diagram of size n×s. An explicit non-zero SL(n,K)-invariant of ∇(λ)⊗∇(λ̃) is calculated; this gives rise to an SL(n,K)-homomorphism φ : ∇(λ̃)∗ → ∇(λ), and the image of φ is L(λ). We show that the spanning set produced in this way is the same, up to sign, as our first spanning set B. For our third method, let R̂(T ) denote the sum of bideterminants corresponding to tableaux S which are row equivalent to T . Let A be the set of R̂(T ) where T is semistandard. Using the Schur algebra, we show that A is a spanning set for L(λ). We show that A is related to B by the Désarménien matrix Ω [D], [G, p. 70]. It is known that ∇(λ) can be defined over Z, in the sense that there is a GL(n,Z)-module ∇Z(λ) which is a finitely generated free Z-module, and our GL(n,K)-module ∇(λ) arises from ∇Z(λ) by base change