Kazhdan-Lusztig cells and decomposition numbers

Geck, Meinolf

doi:10.1090/S1088-4165-98-00042-9

Kazhdan-Lusztig cells and decomposition numbers
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by Meinolf Geck

Represent. Theory 2 (1998), 264-277

DOI: https://doi.org/10.1090/S1088-4165-98-00042-9

Published electronically: June 15, 1998

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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.

References

S. Ariki, On the decomposition numbers of the Hecke algebra of $G(m,1,n)$, J. Math. Kyoto Univ. 36 (1996), 789–808.
C. W. Curtis, Corrections and additions to: “On the degrees and rationality of certain characters of finite Chevalley groups” (Trans. Amer. Math. Soc. 165 (1972), 251–273) by C. T. Benson and Curtis, Trans. Amer. Math. Soc. 202 (1975), 405–406. MR 364483, DOI 10.1090/S0002-9947-1975-0364483-8
Charles W. Curtis and Irving Reiner, Methods of representation theory. Vol. I, Wiley Classics Library, John Wiley & Sons, Inc., New York, 1990. With applications to finite groups and orders; Reprint of the 1981 original; A Wiley-Interscience Publication. MR 1038525
Richard Dipper, Polynomial representations of finite general linear groups in nondescribing characteristic, Representation theory of finite groups and finite-dimensional algebras (Bielefeld, 1991) Progr. Math., vol. 95, Birkhäuser, Basel, 1991, pp. 343–370. MR 1112168, DOI 10.1007/978-3-0348-8658-1_{1}4
Richard Dipper and Gordon James, Representations of Hecke algebras of general linear groups, Proc. London Math. Soc. (3) 52 (1986), no. 1, 20–52. MR 812444, DOI 10.1112/plms/s3-52.1.20
Richard Dipper, Gordon James, and Eugene Murphy, Hecke algebras of type $B_n$ at roots of unity, Proc. London Math. Soc. (3) 70 (1995), no. 3, 505–528. MR 1317512, DOI 10.1112/plms/s3-70.3.505
Peter Fleischmann, On Lusztig’s isomorphism theorems for Hecke algebras and representation theory, J. London Math. Soc. (2) 47 (1993), no. 3, 450–472. MR 1214908, DOI 10.1112/jlms/s2-47.3.450
Paul Fong and Bhama Srinivasan, Brauer trees in classical groups, J. Algebra 131 (1990), no. 1, 179–225. MR 1055005, DOI 10.1016/0021-8693(90)90172-K
Meinolf Geck, Brauer trees of Hecke algebras, Comm. Algebra 20 (1992), no. 10, 2937–2973. MR 1179271, DOI 10.1080/00927879208824499
—, Representations of Hecke algebras at roots of unity, Séminaire Bourbaki, 50ème annèe, 1997-98, Exp. 836, Astérisque (to appear).
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, DOI 10.1007/978-1-4612-4124-9_{9}
Akihiko Gyoja, Modular representation theory over a ring of higher dimension with applications to Hecke algebras, J. Algebra 174 (1995), no. 2, 553–572. MR 1334224, DOI 10.1006/jabr.1995.1139
Akihiko Gyoja, Cells and modular representations of Hecke algebras, Osaka J. Math. 33 (1996), no. 2, 307–341. MR 1416051
David Kazhdan and George Lusztig, Representations of Coxeter groups and Hecke algebras, Invent. Math. 53 (1979), no. 2, 165–184. MR 560412, DOI 10.1007/BF01390031
Alain Lascoux, Bernard Leclerc, and Jean-Yves Thibon, Hecke algebras at roots of unity and crystal bases of quantum affine algebras, Comm. Math. Phys. 181 (1996), no. 1, 205–263. MR 1410572, DOI 10.1007/BF02101678
George Lusztig, On a theorem of Benson and Curtis, J. Algebra 71 (1981), no. 2, 490–498. MR 630610, DOI 10.1016/0021-8693(81)90188-5
George Lusztig, Characters of reductive groups over a finite field, Annals of Mathematics Studies, vol. 107, Princeton University Press, Princeton, NJ, 1984. MR 742472, DOI 10.1515/9781400881772
George Lusztig, Cells in affine Weyl groups, Algebraic groups and related topics (Kyoto/Nagoya, 1983) Adv. Stud. Pure Math., vol. 6, North-Holland, Amsterdam, 1985, pp. 255–287. MR 803338, DOI 10.2969/aspm/00610255
George Lusztig, Cells in affine Weyl groups. II, J. Algebra 109 (1987), no. 2, 536–548. MR 902967, DOI 10.1016/0021-8693(87)90154-2
George Lusztig, Cells in affine Weyl groups. II, J. Algebra 109 (1987), no. 2, 536–548. MR 902967, DOI 10.1016/0021-8693(87)90154-2
G. Lusztig, Leading coefficients of character values of Hecke algebras, The Arcata Conference on Representations of Finite Groups (Arcata, Calif., 1986) Proc. Sympos. Pure Math., vol. 47, Amer. Math. Soc., Providence, RI, 1987, pp. 235–262. MR 933415, DOI 10.1090/pspum/047.2/933415
A. Mathas, Canonical bases and the decomposition matrices of Ariki–Koike algebras, preprint (1996).
J. Müller, Zerlegungszahlen für generische Iwahori–Hecke–Algebren von exzeptionellem Typ, Dissertation, RWTH Aachen, 1995.
Jean-Luc Brylinski, (Co)-homologie d’intersection et faisceaux pervers, Bourbaki Seminar, Vol. 1981/1982, Astérisque, vol. 92, Soc. Math. France, Paris, 1982, pp. 129–157 (French). MR 689529, DOI 10.1090/S0002-9947-1972-0304473-1