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Representation Theory

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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
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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  • 1. N. Bourbaki, Éléments de mathématique. Fasc. XXXIV. Groupes et algèbres de Lie. Chapitre IV: Groupes de Coxeter et systèmes de Tits. Chapitre V: Groupes engendrés par des réflexions. Chapitre VI: systèmes de racines, Actualités Scientifiques et Industrielles, No. 1337, Hermann, Paris, 1968 (French). MR 0240238
  • 2. Michel Broué and Gunter Malle, Zyklotomische Heckealgebren, Astérisque 212 (1993), 119–189 (German). Représentations unipotentes génériques et blocs des groupes réductifs finis. MR 1235834
  • 3. Michel Broué, Gunter Malle, and Jean Michel, Generic blocks of finite reductive groups, Astérisque 212 (1993), 7–92. Représentations unipotentes génériques et blocs des groupes réductifs finis. MR 1235832
  • 4. Michel Broué and Jean Michel, Blocs à groupes de défaut abéliens des groupes réductifs finis, Astérisque 212 (1993), 93–117 (French). Représentations unipotentes génériques et blocs des groupes réductifs finis. MR 1235833
  • 5. Marc Cabanes and Michel Enguehard, Representation theory of finite reductive groups, New Mathematical Monographs, vol. 1, Cambridge University Press, Cambridge, 2004. MR 2057756
  • 6. Charles W. Curtis and Irving Reiner, Methods of representation theory. Vol. I, John Wiley & Sons, Inc., New York, 1981. With applications to finite groups and orders; Pure and Applied Mathematics; A Wiley-Interscience Publication. MR 632548
  • 7. Charles W. Curtis and Irving Reiner, Methods of representation theory. Vol. II, Pure and Applied Mathematics (New York), John Wiley & Sons, Inc., New York, 1987. With applications to finite groups and orders; A Wiley-Interscience Publication. MR 892316
  • 8. Vinay V. Deodhar, Some characterizations of Bruhat ordering on a Coxeter group and determination of the relative Möbius function, Invent. Math. 39 (1977), no. 2, 187–198. MR 0435249
  • 9. Vinay V. Deodhar, On some geometric aspects of Bruhat orderings. I. A finer decomposition of Bruhat cells, Invent. Math. 79 (1985), no. 3, 499–511. MR 782232, 10.1007/BF01388520
  • 10. Vinay V. Deodhar, On some geometric aspects of Bruhat orderings. II. The parabolic analogue of Kazhdan-Lusztig polynomials, J. Algebra 111 (1987), no. 2, 483–506. MR 916182, 10.1016/0021-8693(87)90232-8
  • 11. Meinolf Geck and Götz Pfeiffer, Characters of finite Coxeter groups and Iwahori-Hecke algebras, London Mathematical Society Monographs. New Series, vol. 21, The Clarendon Press, Oxford University Press, New York, 2000. MR 1778802
  • 12. J. J. Graham and G. I. Lehrer, Cellular algebras, Invent. Math. 123 (1996), no. 1, 1–34. MR 1376244, 10.1007/BF01232365
  • 13. David Kazhdan and George Lusztig, Representations of Coxeter groups and Hecke algebras, Invent. Math. 53 (1979), no. 2, 165–184. MR 560412, 10.1007/BF01390031
  • 14. George Lusztig, Characters of reductive groups over a finite field, Annals of Mathematics Studies, vol. 107, Princeton University Press, Princeton, NJ, 1984. MR 742472
  • 15. G. Lusztig, Hecke algebras with unequal parameters, CRM Monograph Series, vol. 18, American Mathematical Society, Providence, RI, 2003. MR 1974442
  • 16. Takeo Yokonuma, Sur la structure des anneaux de Hecke d’un groupe de Chevalley fini, C. R. Acad. Sci. Paris Sér. A-B 264 (1967), A344–A347 (French). MR 0218467

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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

DOI: http://dx.doi.org/10.1090/S1088-4165-2010-00394-5
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.