Coefficient Space Properties and a Schur Algebra Generalization

Abstract

Let K be an infinite field and Gamma = GL_(n)(K). If we linearly extend the natural action of Gamma on the set E of n-dimensional column vectors over K to the group algebra KGamma, then E becomes a KGamma-module. We then construct the KGamma-module E^(otimes r), the r-fold tensor product of E. The image S_(r)(Gamma) of the corresponding representation of KGamma is called the Schur algebra. If E is replaced by a different KGamma-module L, the same construction results in an algebra S_(r,\,L). The subalgebra A(n) of K^(Gamma) generated by the coordinate functions c_(alpha beta) from Gamma to K with 1 <= alpha, beta <= n is a bialgebra. A(n) has a subcoalgebra A_(r) which consists of homogeneous polynomials of total degree r in the indeterminants c_(alpha beta). Classically, the dual A_(r)^* of A_(r) is an algebra isomorphic to S_(r)(Gamma) and A_(r) is the coefficient space of E^(otimes r). We identify S_(r, L) with the dual A_(r, L)^* of the coefficient space A_(r, L) of L^(otimes r) and give a description of A_(r, L).