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Generic Hecke algebras for monomial groups


Authors: S. I. Alhaddad and J. Matthew Douglass
Journal: Represent. Theory 14 (2010), 688-712
MSC (2010): Primary 20C08; Secondary 20F55
DOI: https://doi.org/10.1090/S1088-4165-2010-00394-5
Published electronically: November 15, 2010
MathSciNet review: 2738584
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Abstract: In this paper we define a two-variable, generic Hecke algebra, $\mathcal H$, for each complex reflection group $G(b,1,n)$. The algebra $\mathcal H$ specializes to the group algebra of $G(b,1,n)$ and also to an endomorphism algebra of a representation of $\operatorname {GL}_n(\mathbb F_q)$ induced from a solvable subgroup. We construct Kazhdan-Lusztig “$R$-polynomials” for $\mathcal {H}$ and show that they may be used to define a partial order on $G(b,1,n)$. Using a generalization of Deodhar’s notion of distinguished subexpressions we give a closed formula for the $R$-polynomials. After passing to a one-variable quotient of the ring of scalars, we construct Kazhdan-Lusztig polynomials for $\mathcal H$ that reduce to the usual Kazhdan-Lusztig polynomials for the symmetric group when $b=1$.


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Additional Information

S. I. Alhaddad
Affiliation: Department of Mathematics, University of South Carolina, Lancaster, Lancaster, South Carolina 29721
Email: alhaddad@gwm.sc.edu

J. Matthew Douglass
Affiliation: Department of Mathematics, University of North Texas, 1155 Union Circle #311430, Denton, Texas 76203-5017
Email: douglass@unt.edu

Received by editor(s): October 8, 2007
Received by editor(s) in revised form: September 17, 2010, and September 25, 2010
Published electronically: November 15, 2010
Additional Notes: The authors would like to thank Nathaniel Thiem for helpful discussions.
Article copyright: © Copyright 2010 American Mathematical Society
The copyright for this article reverts to public domain 28 years after publication.