Monomial bases for $q$-Schur algebras

Abstract

Using the Beilinson-Lusztig-MacPherson construction of the quantized enveloping algebra of gl n and its associated monomial basis, we investigate q-Schur algebras S q (n,r) as little quantum groups. We give a presentation for S q (n,r) and obtain a new basis for the integral q-Schur algebra S q (n,r), which consists of certain monomials in the original generators. Finally, when n ≥ r, we interpret the Hecke algebra part of the monomial basis for S q (n,r) in terms of Kazhdan-Lusztig basis elements.