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Transactions of the American Mathematical Society

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Monomial bases for $q$-Schur algebras

Authors: Jie Du and Brian Parshall
Journal: Trans. Amer. Math. Soc. 355 (2003), 1593-1620
MSC (2000): Primary 17B37, 20C08, 20G05
Published electronically: November 14, 2002
MathSciNet review: 1946407
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Abstract: Using the Beilinson-Lusztig-MacPherson construction of the quantized enveloping algebra of $\mathfrak{gl}_n$ and its associated monomial basis, we investigate $q$-Schur algebras $\mathbf{S}_q(n,r)$ as ``little quantum groups". We give a presentation for $\mathbf{S}_q(n,r)$ 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 $n\geqslant r$, we interpret the Hecke algebra part of the monomial basis for $S_q(n,r)$ in terms of Kazhdan-Lusztig basis elements.

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Additional Information

Jie Du
Affiliation: School of Mathematics, University of New South Wales, Sydney 2052, Australia

Brian Parshall
Affiliation: Department of Mathematics, University of Virginia, Charlottesville, Virginia 22904-4137

Received by editor(s): October 1, 2001
Received by editor(s) in revised form: July 1, 2002
Published electronically: November 14, 2002
Additional Notes: Supported partially by ARC and NSF
Article copyright: © Copyright 2002 American Mathematical Society