## 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 βlittle 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