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Centralizers of Iwahori-Hecke algebras


Author: Andrew Francis
Journal: Trans. Amer. Math. Soc. 353 (2001), 2725-2739
MSC (2000): Primary 20C33, 20F55
DOI: https://doi.org/10.1090/S0002-9947-01-02693-9
Published electronically: March 2, 2001
MathSciNet review: 1828470
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Abstract:

To date, integral bases for the centre of the Iwahori-Hecke algebra of a finite Coxeter group have relied on character theoretical results and the isomorphism between the Iwahori-Hecke algebra when semisimple and the group algebra of the finite Coxeter group. In this paper, we generalize the minimal basis approach of an earlier paper, to provide a way of describing and calculating elements of the minimal basis for the centre of an Iwahori-Hecke algebra which is entirely combinatorial in nature, and independent of both the above mentioned theories.

This opens the door to further generalization of the minimal basis approach to other cases. In particular, we show that generalizing it to centralizers of parabolic subalgebras requires only certain properties in the Coxeter group. We show here that these properties hold for groups of type $A$ and $B$, giving us the minimal basis theory for centralizers of any parabolic subalgebra in these types of Iwahori-Hecke algebra.


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

  • [C] R. Carter, Representation theory of the $0$-Hecke algebra, J. Algebra 104 (1986), 89-103. MR 88a:20014
  • [DD] R. Dipper and J. Du, Trivial and alternating source modules of Hecke algebras of type $A$, Proc. London Math. Soc. 66 (3) (1993), 479-506. MR 94a:20022
  • [F1] A. Francis, The minimal basis for the centre of an Iwahori-Hecke algebra, J. Algebra 221 (1999), 1-28. MR 2000k:20005
  • [F2] -, Centralizers of Iwahori-Hecke algebras II: the general case, preprint.
  • [GHLMP] M. Geck, G. Hiss, F. Lübeck, G. Malle, and G. Pfeiffer, CHEVIE--a system for computing and processing generic character tables. Computational methods in Lie theory (Essen 1994), Appl. Algebra Engrg. Comm. Comput. 7 (3) (1996), 175-210. MR 99m:20017
  • [GP] M. Geck and G. Pfeiffer, On the irreducible characters of Iwahori-Hecke algebras, Adv. Math. 102 (1993), 79-94. MR 94m:20018
  • [GR] M. Geck and R. Rouquier, Centers and simple modules for Iwahori-Hecke algebras, in Finite Reductive Groups (Luminy, 1994), Prog. Math., vol. 141, Birkhäuser, Boston, MA, 1997, pp. 251-272. MR 98c:20013
  • [J] L. Jones, Centers of generic Hecke algebras, Trans. Amer. Math. Soc. 317 (1990), 361-392. MR 90d:20030
  • [R] A. Ram, A Frobenius formula for the characters of the Hecke algebras, Invent. Math 106 (1991), 461-488. MR 93c:20029

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Additional Information

Andrew Francis
Affiliation: Department of Mathematics, University of Virginia, Charlottesville, Virginia 22903
Address at time of publication: University of Western Sydney, Richmond, NSW 2753, Australia
Email: a.francis@uws.edu.au

DOI: https://doi.org/10.1090/S0002-9947-01-02693-9
Keywords: Coxeter groups, Iwahori-Hecke algebras, centre, centralizer, minimal basis
Received by editor(s): October 8, 1998
Received by editor(s) in revised form: December 21, 1999
Published electronically: March 2, 2001
Additional Notes: The diagrams in this paper were created using Paul Taylor’s Commutative Diagrams package. The research for this paper was in part supported by an Australian Postgraduate Award, and was done partially as part of work towards a Ph.D. at the University of New South Wales
Article copyright: © Copyright 2001 American Mathematical Society

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