Presenting generalized 𝑞-Schur algebras

Abstract

We obtain a presentation by generators and relations for generalized Schur algebras and their quantizations. This extends earlier results obtained in the type A A case. The presentation is compatible with Lusztig’s modified form U ˙ \mathbf {\dot {U}} of a quantized enveloping algebra. We show that generalized Schur algebras inherit a canonical basis from U ˙ \mathbf {\dot {U}} , that this gives them a cellular structure, and thus they are quasihereditary over a field.