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Transactions of the American Mathematical Society

Published by the American Mathematical Society, the Transactions of the American Mathematical Society (TRAN) is devoted to research articles of the highest quality in all areas of pure and applied mathematics.

ISSN 1088-6850 (online) ISSN 0002-9947 (print)

The 2020 MCQ for Transactions of the American Mathematical Society is 1.43.

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Monomial bases for $q$-Schur algebras
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by Jie Du and Brian Parshall PDF
Trans. Amer. Math. Soc. 355 (2003), 1593-1620 Request permission

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 β€œlittle 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.
References
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Additional Information
  • Jie Du
  • Affiliation: School of Mathematics, University of New South Wales, Sydney 2052, Australia
  • MR Author ID: 242577
  • Email: j.du@unsw.edu.au
  • Brian Parshall
  • Affiliation: Department of Mathematics, University of Virginia, Charlottesville, Virginia 22904-4137
  • MR Author ID: 136395
  • Email: bjp8w@virginia.edu
  • 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
  • © Copyright 2002 American Mathematical Society
  • Journal: Trans. Amer. Math. Soc. 355 (2003), 1593-1620
  • MSC (2000): Primary 17B37, 20C08, 20G05
  • DOI: https://doi.org/10.1090/S0002-9947-02-03188-4
  • MathSciNet review: 1946407