Schur algebras

(1.2) Let k be an algebraically closed field of arbitrary characteristic. For a ring A, an A-module means a left A-module, unless otherwise specified. However, an ideal of A means a two-sided ideal, not a left ideal. Amod denotes the category of finitely generated A-modules. For a group G, a G-module means a kG-module, where kG is the group algebra of G over k. If V is a finite dimensional vector space, then giving a G-module structure to V is the same thing as giving a group homomorphism ρ : G → GL(V ). A finite dimensional GLn(k)-module V ∼= k m is said to be a polynomial (resp. rational) representation if the corresponding group homomorphism ρ : GLn(k) → GL(V ) ∼= GLm(k) satisfies the following. For each (aij) ∈ GLn(k), when we write ρ(aij) = (ρst(aij)), then each ρst(aij) is a polynomial function (resp. rational function everywhere defined on GLn) in aij . We may also say that ρ is a polynomial (resp. rational) representation. Note that this condition is independent of the choice of the basis of V .