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Representation Theory

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Kazhdan-Lusztig cells and decomposition numbers

Author: Meinolf Geck
Journal: Represent. Theory 2 (1998), 264-277
MSC (1991): Primary 20C20; Secondary 20G99
Published electronically: June 15, 1998
MathSciNet review: 1628035
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Abstract: We consider a generic Iwahori-Hecke algebra $H$ associated with a finite Weyl group. Any specialization of $H$ gives rise to a corresponding decomposition matrix, and we show that the problem of computing that matrix can be interpreted in terms of Lusztig's map from $H$ to the asymptotic algebra $J$. This interpretation allows us to prove that the decomposition matrices always have a lower uni-triangular shape; moreover, we determine these matrices explicitly in the so-called defect $1$ case.

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  • 1. S. Ariki, On the decomposition numbers of the Hecke algebra of $G(m,1,n)$, J. Math. Kyoto Univ. 36 (1996), 789-808. CMP 97:11
  • 2. C. T. Benson and C. W. Curtis, On the degrees and rationality of certain characters of finite Chevalley groups, Trans. Amer. Math. Soc. 165 (1972), 251-273; Corrections and additions, ibid. 202 (1975), 405-406. MR 51:737
  • 3. C. W. Curtis and I. Reiner, Methods of Representation Theory Vols. I and II, Wiley, New York, 1981 and 1987. MR 90k:20001; MR 82i:20001; MR 88f:20002
  • 4. R. Dipper, Polynomial representations of finite general linear groups in non-describing characteristic, Representation theory of finite groups and finite-dimensional algebras (G. O. Michler, C. M. Ringel, eds.), Progress in Math. 95, Birkhäuser, Boston, 1991, pp. 344-370. MR 92h:20018
  • 5. R. Dipper and G. D. James, Representations of Hecke algebras of the general linear groups, Proc. London Math. Soc. 52 (1986), 20-52. MR 88b:20065
  • 6. R. Dipper, G. D. James and G. E. Murphy, Hecke algebras of type $B_n$ at roots of unity, Proc. London Math. Soc. 70 (1995), 505-528. MR 96b:20004
  • 7. P. Fleischmann, On Lusztig's isomorphism theorems for Hecke algebras and representation theory, J. London Math. Soc. 47 (1993), 450-472. MR 94c:20074
  • 8. P. Fong and B. Srinivasan, Brauer trees in classical groups, J. Algebra 131 (1990), 179-225. MR 91i:20008
  • 9. M. Geck, Brauer trees of Hecke algebras, Comm. Algebra 20 (1992), 2937-2973. MR 94a:20019
  • 10. -, Representations of Hecke algebras at roots of unity, Séminaire Bourbaki, 50ème annèe, 1997-98, Exp. 836, Astérisque (to appear).
  • 11. M. Geck and R. Rouquier, Centers and simple modules for Iwahori-Hecke algebras, Finite reductive groups, related structures and representations (M. Cabanes, ed.), Progress in Math. 141, Birkhäuser, Boston, 1997, pp. 251-272. MR 98c:20013
  • 12. A. Gyoja, Modular representation theory over a ring of higher dimension with applications to Hecke algebras, J. Algebra 174 (1995), 553-572. MR 96m:20024
  • 13. -, Cells and modular representations of Hecke algebras, Osaka J. Math. 33 (1996), 307-341. MR 97k:20018
  • 14. D. Kazhdan and G. Lusztig, Representations of Coxeter groups and Hecke algebras, Invent. Math. 53 (1979), 165-184. MR 81j:20066
  • 15. A. Lascoux, B. Leclerc and J. Y. Thibon, Hecke algebras at roots of unity and crystal bases of quantum affine algebras, Comm. Math. Physics 181 (1996), 205-263. MR 97k:17019
  • 16. G. Lusztig, On a theorem of Benson and Curtis, J. Algebra 71 (1981), 490-498. MR 83a:20053
  • 17. -, Characters of reductive groups over a finite field, Annals Math. Studies 107, Princeton University Press, 1984. MR 86j:20038
  • 18. -, Cells in affine Weyl groups, Advanced Studies in Pure Math. 6, Algebraic groups and related topics, Kinokuniya and North-Holland, Tokyo-Amsterdam, 1985, pp. 255-287. MR 87h:20074
  • 19. -, Cells in affine Weyl groups II, J. Algebra 109 (1987), 536-548. MR 88m:20103a
  • 20. -, Cells in affine Weyl groups III, J. Fac. Sci. Univ. Tokyo 34 (1987), 223-243. MR 88m:20103b
  • 21. -, Leading coefficients of character values of Hecke algebra, Proc. Symp. Pure Math. 47, Amer. Math. Soc., Providence, RI, 1987, pp. 235-262. MR 89b:20087
  • 22. A. Mathas, Canonical bases and the decomposition matrices of Ariki-Koike algebras, preprint (1996).
  • 23. J. Müller, Zerlegungszahlen für generische Iwahori-Hecke-Algebren von exzeptionellem Typ, Dissertation, RWTH Aachen, 1995.
  • 24. T. A. Springer, Quelques applications de la cohomologie d'intersection, Séminare Bourbaki (1981/82), exp. 589, Astérisque 92-93 (1982). MR 85i:32016b

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Additional Information

Meinolf Geck
Affiliation: UFR de Mathématiques and UMR 7586 du CNRS, Université Paris 7, 2 Place Jussieu, 75251 Paris, France

Received by editor(s): December 11, 1997
Received by editor(s) in revised form: April 27, 1998
Published electronically: June 15, 1998
Article copyright: © Copyright 1998 American Mathematical Society

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