Remote Access Transactions of the American Mathematical Society
Green Open Access

Transactions of the American Mathematical Society

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



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
Full-text PDF

Abstract | References | Similar Articles | Additional Information

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.

References [Enhancements On Off] (What's this?)

  • 1. A. A. Beilinson, G. Lusztig and R. MacPherson, A geometric setting for the quantum deformation of $\operatorname{GL}_n$, Duke Math. J. 61 (1990), 655-677. MR 91m:17012
  • 2. K. Bongartz, On degenerations and extensions of finite-dimensional modules, Adv. Math. 121 (1996), 245-287. MR 98e:16012
  • 3. V. Chari and N. Xi, Monomial bases of quantized enveloping algebras, in ``Recent developments in quantum affine algebras and related topics'' (Raleigh, NC, 1998), 69-81, Contemp. Math., 248, Amer. Math. Soc., Providence, RI, 1999. MR 2001c:17023
  • 4. B. Deng and J. Du, On bases of quantized enveloping algebras, to appear.
  • 5. R. Dipper and G. James, The $q$-Schur algebra, Proc. London Math. Soc. 59 (1989), 23-50. MR 90g:16026
  • 6. R. Dipper and G. James, $q$-tensor space and $q$-Weyl modules, Trans. Amer. Math. Soc. 327 (1991), 251-282. MR 91m:20061
  • 7. S. Doty and A. Giaquinto, Presenting quantum Schur algebras as quotients of the quantized universal enveloping algebras of ${\mathfrak{gl}}_2$, preprint.
  • 8. S. Doty and A. Giaquinto, Presenting Schur algebras, International Mathematics Research Notices IMRN 2002:36 (2002) 1907-1944.
  • 9. J. Du, A note on the quantized Weyl reciprocity at roots of unity, Alg. Colloq. 2 (1995), 363-372. MR 96m:17024
  • 10. J. Du, $q$-Schur algebras, asymptotic forms, and quantum $SL_n$, J. Algebra 177 (1995), 385-408. MR 96k:17021
  • 11. J. Du, B. Parshall and L. Scott, Quantum Weyl reciprocity and tilting modules, Comm. Math. Phys. 195 (1998), 321-352. MR 99k:17026
  • 12. J. Du and B. Parshall, Linear quivers and the geometric setting for quantum $GL_n$, Indag. Math., in press.
  • 13. J. Du and H. Rui, Based algebras and standard bases for quasi-hereditary algebras, Trans. Amer. Math. Soc. 350 (1998), 3207-3235. MR 99b:16027
  • 14. R. Green, $q$-Schur algebras as quotients of quantized enveloping algebras, J. Algebra 185 (1996), 660-687. MR 97k:17016
  • 15. D. Kazhdan and G. Lusztig, Representations of Coxeter groups and Hecke algebras, Invent. Math. 53 (1979), 165-184. MR 81j:20066
  • 16. M. Jimbo, A $q$-analogue of $U(gl(N+1))$, Hecke algebras, and the Yang-Baxter equation, Lett. Math. Phy. 11 (1986), 247-252. MR 87k:17011
  • 17. G. Lusztig, Modular representations and quantum groups, Contemp. Math. 82 (1989) 59-77. MR 90a:16008
  • 18. G. Lusztig, Finite-dimensional Hopf algebras arising from quantized universal enveloping algebras, J. Amer. Math. Soc. 3 (1990), 257-296. MR 91e:17009
  • 19. G. Lusztig, Canonical bases arising from quantized enveloping algebras, J. Amer. Math. Soc. 3 (1990), 447-498. MR 90m:17023
  • 20. G. Lusztig, Aperiodicity in quantum affine ${\mathfrak{gl}}_n$, Asian J. Math. 3 (1999), 147-177. MR 2000i:17027
  • 21. G. Lusztig, Transfer maps for quantum affine ${\mathfrak{sl}}_n$, In Representations and quantizations (Shanghai, 1998), China High. Educ. Press, Beijing (2000), 341-356. MR 2002f:17026
  • 22. O. Schiffmann and E. Vasserot, Geometric construction of the global base of the quantum modified algebra of $\hat {\mathfrak{gl}}_n$, Transf. Groups 5 (2000), 351-360. MR 2001k:17029
  • 23. M. Takeuchi, Some topics on $GL_q(n)$, J. Algebra 147 (1992), 379-410. MR 93b:17055
  • 24. H. Wenzl, Hecke algebra of type $A_n$ and subfactors, Invent. Math. 92 (1988), 349-383. MR 90b:46118

Similar Articles

Retrieve articles in Transactions of the American Mathematical Society with MSC (2000): 17B37, 20C08, 20G05

Retrieve articles in all journals with MSC (2000): 17B37, 20C08, 20G05

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

American Mathematical Society