Monomial bases for q-Schur algebras

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 greater than or equal to r, we interpret the Hecke algebra part of the monomial basis for S-q(n, r) in terms of Kazhdan-Lusztig basis elements.