Centralizers of Iwahori-Hecke algebras

Author:
Andrew Francis

Journal:
Trans. Amer. Math. Soc. **353** (2001), 2725-2739

MSC (2000):
Primary 20C33, 20F55

Published electronically:
March 2, 2001

MathSciNet review:
1828470

Full-text PDF Free Access

Abstract | References | Similar Articles | Additional Information

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 and , giving us the minimal basis theory for centralizers of any parabolic subalgebra in these types of Iwahori-Hecke algebra.

**[C]**R. W. Carter,*Representation theory of the 0-Hecke algebra*, J. Algebra**104**(1986), no. 1, 89–103. MR**865891**, 10.1016/0021-8693(86)90238-3**[DD]**Richard Dipper and Jie Du,*Trivial and alternating source modules of Hecke algebras of type 𝐴*, Proc. London Math. Soc. (3)**66**(1993), no. 3, 479–506. MR**1207545**, 10.1112/plms/s3-66.3.479**[F1]**Andrew Francis,*The minimal basis for the centre of an Iwahori-Hecke algebra*, J. Algebra**221**(1999), no. 1, 1–28. MR**1722901**, 10.1006/jabr.1998.7925**[F2]**-,*Centralizers of Iwahori-Hecke algebras*II:*the general case*, preprint.**[GHLMP]**Meinolf Geck, Gerhard Hiss, Frank Lübeck, Gunter Malle, and Götz Pfeiffer,*CHEVIE—a system for computing and processing generic character tables*, Appl. Algebra Engrg. Comm. Comput.**7**(1996), no. 3, 175–210. Computational methods in Lie theory (Essen, 1994). MR**1486215**, 10.1007/BF01190329**[GP]**Meinolf Geck and Götz Pfeiffer,*On the irreducible characters of Hecke algebras*, Adv. Math.**102**(1993), no. 1, 79–94. MR**1250466**, 10.1006/aima.1993.1056**[GR]**Meinolf Geck and Raphaël Rouquier,*Centers and simple modules for Iwahori-Hecke algebras*, Finite reductive groups (Luminy, 1994) Progr. Math., vol. 141, Birkhäuser Boston, Boston, MA, 1997, pp. 251–272. MR**1429875****[J]**Lenny K. Jones,*Centers of generic Hecke algebras*, Trans. Amer. Math. Soc.**317**(1990), no. 1, 361–392. MR**948191**, 10.1090/S0002-9947-1990-0948191-6**[R]**Arun Ram,*A Frobenius formula for the characters of the Hecke algebras*, Invent. Math.**106**(1991), no. 3, 461–488. MR**1134480**, 10.1007/BF01243921

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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