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Conjugacy classes of involutions and Kazhdan-Lusztig cells


Authors: Cédric Bonnafé and Meinolf Geck
Journal: Represent. Theory 18 (2014), 155-182
MSC (2000): Primary 20C08; Secondary 20F55
DOI: https://doi.org/10.1090/S1088-4165-2014-00456-4
Published electronically: July 22, 2014
MathSciNet review: 3233059
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Abstract: According to an old result of Schützenberger, the involutions in a given two-sided cell of the symmetric group $ \mathfrak{S}_n$ are all conjugate. In this paper, we study possible generalizations of this property to other types of Coxeter groups. We show that Schützenberger's result is a special case of a general result on ``smooth'' two-sided cells. Furthermore, we consider Kottwitz's conjecture concerning the intersections of conjugacy classes of involutions with the left cells in a finite Coxeter group. Our methods lead to a proof of this conjecture for classical types which, combined with further recent work, settles this conjecture in general.


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  • [Al] Dean Alvis, The left cells of the Coxeter group of type $ H_4$, J. Algebra 107 (1987), no. 1, 160-168. MR 883878 (88d:20014), https://doi.org/10.1016/0021-8693(87)90082-2
  • [Bon1] Cédric Bonnafé, Two-sided cells in type $ B$ (asymptotic case), J. Algebra 304 (2006), no. 1, 216-236. MR 2255826 (2007j:20003), https://doi.org/10.1016/j.jalgebra.2006.05.035
  • [BGIL] Cédric Bonnafé, Meinolf Geck, Lacrimioara Iancu, and Thomas Lam, On domino insertion and Kazhdan-Lusztig cells in type $ B_n$, Representation theory of algebraic groups and quantum groups, Progr. Math., vol. 284, Birkhäuser/Springer, New York, 2010, pp. 33-54. MR 2761947 (2011k:20004), https://doi.org/10.1007/978-0-8176-4697-4_3
  • [BoIa] Cédric Bonnafé and Lacrimioara Iancu, Left cells in type $ B_n$ with unequal parameters, Represent. Theory 7 (2003), 587-609 (electronic). MR 2017068 (2004j:20007), https://doi.org/10.1090/S1088-4165-03-00188-2
  • [BoRo1] Cédric Bonnafé and Raphaël Rouquier, Calogero-Moser versus Kazhdan-Lusztig cells, Pacific J. Math. 261 (2013), no. 1, 45-51. MR 3037558, https://doi.org/10.2140/pjm.2013.261.45
  • [BoRo2] Cédrick Bonnafé and Raphaël Rouquier, Cellules de Calogero-Moser; preprint arXiv:1302.2720.
  • [Ca] Bill Casselman, Verifying Kottwitz' conjecture by computer, Represent. Theory 4 (2000), 32-45 (electronic). MR 1740179 (2000k:20059), https://doi.org/10.1090/S1088-4165-00-00052-2
  • [Du] Fokko du Cloux, Positivity results for the Hecke algebras of noncrystallographic finite Coxeter groups, J. Algebra 303 (2006), no. 2, 731-741. MR 2255133 (2007e:20010), https://doi.org/10.1016/j.jalgebra.2005.10.004
  • [Ge1] Meinolf Geck, Computing Kazhdan-Lusztig cells for unequal parameters, J. Algebra 281 (2004), no. 1, 342-365. MR 2091976 (2005g:20007), https://doi.org/10.1016/j.jalgebra.2004.07.029
  • [Ge2] Meinolf Geck, Left cells and constructible representations, Represent. Theory 9 (2005), 385-416 (electronic). MR 2133765 (2005m:20017), https://doi.org/10.1090/S1088-4165-05-00245-1
  • [Ge3] Meinolf Geck, Kazhdan-Lusztig cells and the Murphy basis, Proc. London Math. Soc. (3) 93 (2006), no. 3, 635-665. MR 2266962 (2008f:20012), https://doi.org/10.1017/S0024611506015930
  • [Ge4] Meinolf Geck, On Iwahori-Hecke algebras with unequal parameters and Lusztig's isomorphism theorem, Pure Appl. Math. Q. 7 (2011), no. 3, Special Issue: In honor of Jacques Tits, 587-620. MR 2848588 (2012g:20008), https://doi.org/10.4310/PAMQ.2011.v7.n3.a5
  • [Ge5] Meinolf Geck, Kazhdan-Lusztig cells and the Frobenius-Schur indicator, J. Algebra 398 (2014), 329-342. MR 3123768, https://doi.org/10.1016/j.jalgebra.2013.01.019
  • [Ge6] Meinolf Geck, $ \mathsf {PyCox}$: computing with (finite) Coxeter groups and Iwahori-Hecke algebras, LMS J. Comput. Math. 15 (2012), 231-256. MR 2988815, https://doi.org/10.1112/S1461157012001064
  • [Ge7] Meinolf Geck, On Kottwitz' conjecture for twisted involutions, see arXiv:1206.0443.
  • [GeHa] Meinolf Geck and Abbie Halls, On the Kazhdan-Lusztig cells in type $ E_8$, Mathematics of Computation, to appear.
  • [GeJa] Meinolf Geck and Nicolas Jacon, Representations of Hecke algebras at roots of unity, Algebra and Applications, vol. 15, Springer-Verlag London, Ltd., London, 2011. MR 2799052 (2012d:20010)
  • [GeMa] Meinolf Geck and Gunter Malle, Frobenius-Schur indicators of unipotent characters and the twisted involution module, Represent. Theory 17 (2013), 180-198. MR 3037782, https://doi.org/10.1090/S1088-4165-2013-00430-2
  • [GePf] 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 (2002k:20017)
  • [Gu] Jérémie Guilhot, On the lowest two-sided cell in affine Weyl groups, Represent. Theory 12 (2008), 327-345. MR 2448287 (2010e:20009), https://doi.org/10.1090/S1088-4165-08-00334-8
  • [Ho] Christophe Hohlweg, A generalization of plactic-coplactic equivalences and Kazhdan-Lusztig cells, J. Algebra 283 (2005), no. 2, 671-689. MR 2111216 (2005j:20044), https://doi.org/10.1016/j.jalgebra.2004.09.018
  • [KaLu] David Kazhdan and George Lusztig, Representations of Coxeter groups and Hecke algebras, Invent. Math. 53 (1979), no. 2, 165-184. MR 560412 (81j:20066), https://doi.org/10.1007/BF01390031
  • [Ko] Robert E. Kottwitz, Involutions in Weyl groups, Represent. Theory 4 (2000), 1-15 (electronic). MR 1740177 (2000m:22014), https://doi.org/10.1090/S1088-4165-00-00050-9
  • [Lu1] George Lusztig, Left cells in Weyl groups, Lie group representations, I (College Park, Md., 1982/1983) Lecture Notes in Math., vol. 1024, Springer, Berlin, 1983, pp. 99-111. MR 727851 (85f:20035), https://doi.org/10.1007/BFb0071433
  • [LuB] George Lusztig, Characters of reductive groups over a finite field, Annals of Mathematics Studies, vol. 107, Princeton University Press, Princeton, NJ, 1984. MR 742472 (86j:20038)
  • [Lu2] 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 (87h:20074)
  • [Lu3] George Lusztig, Sur les cellules gauches des groupes de Weyl, C. R. Acad. Sci. Paris Sér. I Math. 302 (1986), no. 1, 5-8 (French, with English summary). MR 827096 (87e:20089)
  • [Lu4] George 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 (89b:20087)
  • [Lu5] George Lusztig, Hecke algebras with unequal parameters, CRM Monograph Series, vol. 18, American Mathematical Society, Providence, RI, 2003. MR 1974442 (2004k:20011)
  • [Lu6] George Lusztig, A bar operator for involutions in a Coxeter group, Bull. Inst. Math. Acad. Sin. (N.S.) 7 (2012), no. 3, 355-404. MR 3051318
  • [LuVo] George Lusztig and David A. Vogan Jr., Hecke algebras and involutions in Weyl groups, Bull. Inst. Math. Acad. Sin. (N.S.) 7 (2012), no. 3, 323-354. MR 3051317
  • [Ma] Eric Marberg, How to compute the Frobenius-Schur indicator of a unipotent character of a finite Coxeter system, Adv. Math. 240 (2013), 484-519. MR 3046317, https://doi.org/10.1016/j.aim.2013.02.023
  • [Sch] Marcel-Paul Schützenberger, La correspondance de Robinson, Combinatoire et représentation du groupe symétrique (Actes Table Ronde CNRS, Univ. Louis-Pasteur Strasbourg, Strasbourg, 1976) Springer, Berlin, 1977, pp. 59-113. Lecture Notes in Math., Vol. 579 (French). MR 0498826 (58 #16863)

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Additional Information

Cédric Bonnafé
Affiliation: Institut de Mathématiques et de Modélisation de Montpellier (CNRS: UMR 5149), Université Montpellier 2, Case Courrier 051, Place Eugène Bataillon, 34095 Montpellier Cedex, France
Email: cedric.bonnafe@math.univ-montp2.fr

Meinolf Geck
Affiliation: IAZ - Lehrstuhl für Algebra, Universität Stuttgart, Pfaffenwaldring 57, D–70569 Stuttgart, Germany
Email: meinolf.geck@mathematik.uni-stuttgart.de

DOI: https://doi.org/10.1090/S1088-4165-2014-00456-4
Received by editor(s): January 28, 2013
Received by editor(s) in revised form: July 1, 2014
Published electronically: July 22, 2014
Article copyright: © Copyright 2014 American Mathematical Society

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