Remote Access Mathematics of Computation
Green Open Access

Mathematics of Computation

ISSN 1088-6842(online) ISSN 0025-5718(print)

Request Permissions   Purchase Content 
 
 

 

On the Kazhdan-Lusztig cells in type $ E_8$


Authors: Meinolf Geck and Abbie Halls
Journal: Math. Comp. 84 (2015), 3029-3049
MSC (2010): Primary 20C40; Secondary 20C08, 20F55
DOI: https://doi.org/10.1090/mcom/2963
Published electronically: May 8, 2015
MathSciNet review: 3378861
Full-text PDF

Abstract | References | Similar Articles | Additional Information

Abstract: In 1979, Kazhdan and Lusztig introduced the notion of ``cells'' (left, right and two-sided) for a Coxeter group $ W$, a concept with numerous applications in Lie theory and around. Here, we address algorithmic aspects of this theory for finite $ W$ which are important in applications, e.g., run explicitly through all left cells, determine the values of Lusztig's $ \mathbf {a}$-function, identify the characters of left cell representations. The aim is to show how type $ E_8$ (the largest group of exceptional type) can be handled systematically and efficiently, too. This allows us, for the first time, to solve some open questions in this case, including Kottwitz' conjecture on left cells and involutions.


References [Enhancements On Off] (What's this?)

  • [1] 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
  • [2] Dean Alvis and George Lusztig, The representations and generic degrees of the Hecke algebra of type $ H_{4}$, J. Reine Angew. Math. 336 (1982), 201-212. MR 671329 (84a:20013), https://doi.org/10.1515/crll.1982.336.201
  • [3] 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
  • [4] Cédric Bonnafé and Meinolf Geck, Conjugacy classes of involutions and Kazhdan-Lusztig cells, Represent. Theory 18 (2014), 155-182. MR 3233059, https://doi.org/10.1090/S1088-4165-2014-00456-4
  • [5] 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
  • [6] Yu Chen, Left cells in the Weyl group of type $ E_8$, J. Algebra 231 (2000), no. 2, 805-830. MR 1778172 (2001k:20081), https://doi.org/10.1006/jabr.1999.8376
  • [7] Yu Chen and Jian-Yi Shi, Left cells in the Weyl group of type $ E_7$, Comm. Algebra 26 (1998), no. 11, 3837-3852. MR 1647091 (99h:20065), https://doi.org/10.1080/00927879208826378
  • [8] Fokko du Cloux, The state of the art in the computation of Kazhdan-Lusztig polynomials, Appl. Algebra Engrg. Comm. Comput. 7 (1996), no. 3, 211-219. Computational methods in Lie theory (Essen, 1994). MR 1486216 (98m:20047), https://doi.org/10.1007/BF01190330
  • [9] 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
  • [10] Ben Elias and Geordie Williamson, The Hodge theory of Soergel bimodules, Ann. of Math. (2) 180 (2014), no. 3, 1089-1136. MR 3245013, https://doi.org/10.4007/annals.2014.180.3.6
  • [11] Devra Garfinkle, On the classification of primitive ideals for complex classical Lie algebras. III, Compositio Math. 88 (1993), no. 2, 187-234. MR 1237920 (94i:17017)
  • [12] Meinolf Geck, On the induction of Kazhdan-Lusztig cells, Bull. London Math. Soc. 35 (2003), no. 5, 608-614. MR 1989489 (2004d:20003), https://doi.org/10.1112/S0024609303002236
  • [13] Meinolf Geck, Constructible characters, leading coefficients and left cells for finite Coxeter groups with unequal parameters, Represent. Theory 6 (2002), 1-30. MR 1915085 (2003d:20009), https://doi.org/10.1090/S1088-4165-02-00128-0
  • [14] Meinolf Geck, Relative Kazhdan-Lusztig cells, Represent. Theory 10 (2006), 481-524 (electronic). MR 2266700 (2007i:20011), https://doi.org/10.1090/S1088-4165-06-00287-1
  • [15] 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
  • [16] Meinolf Geck, Leading coefficients and cellular bases of Hecke algebras, Proc. Edinb. Math. Soc. (2) 52 (2009), no. 3, 653-677. MR 2546637 (2011d:20005), https://doi.org/10.1017/S0013091508000394
  • [17] Meinolf Geck, On Iwahori-Hecke algebras with unequal parameters and Lusztig's isomorphism theorem, Pure Appl. Math. Q. 7 (2011), no. 3, 587-620. MR 2848588 (2012g:20008), https://doi.org/10.4310/PAMQ.2011.v7.n3.a5
  • [18] Meinolf Geck, On the Kazhdan-Lusztig order on cells and families, Comment. Math. Helv. 87 (2012), no. 4, 905-927. MR 2984576, https://doi.org/10.4171/CMH/273
  • [19] 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
  • [20] 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
  • [21] M. Geck, On Kottwitz' conjecture for twisted involutions, J. Lie Theory 25 (2015), 395-429.
  • [22] M. Geck, PyCox - A Python version of CHEVIE-GAP for (finite) Coxeter groups, version 1r6p180, 2014 (http://www.mathematik.uni-stuttgart.de/~geckmf).
  • [23] M. Geck, A generalised $ \tau $-invariant for the unequal parameter case, preprint, available at arXiv:1405.5736 (see also arXiv:1502.01661).
  • [24] 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)
  • [25] Meinolf Geck and Jürgen Müller, James' conjecture for Hecke algebras of exceptional type. I, J. Algebra 321 (2009), no. 11, 3274-3298. MR 2510049 (2010h:20012), https://doi.org/10.1016/j.jalgebra.2008.10.024
  • [26] 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)
  • [27] R. B. Howlett, $ W$-graphs for the irreducible representations of the Hecke algebra of type $ E_8$, private communication with J. Michel (December 2003).
  • [28] Robert B. Howlett and Yunchuan Yin, Computational construction of irreducible $ W$-graphs for types $ E_6$ and $ E_7$, J. Algebra 321 (2009), no. 8, 2055-2067. MR 2501509 (2010b:20004), https://doi.org/10.1016/j.jalgebra.2008.12.018
  • [29] James E. Humphreys, Reflection Groups and Coxeter Groups, Cambridge Studies in Advanced Mathematics, vol. 29, Cambridge University Press, Cambridge, 1990. MR 1066460 (92h:20002)
  • [30] 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
  • [31] 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
  • [32] George Lusztig, On a theorem of Benson and Curtis, J. Algebra 71 (1981), no. 2, 490-498. MR 630610 (83a:20053), https://doi.org/10.1016/0021-8693(81)90188-5
  • [33] 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)
  • [34] 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)
  • [35] George Lusztig, Cells in affine Weyl groups. II, J. Algebra 109 (1987), no. 2, 536-548. MR 902967 (88m:20103a), https://doi.org/10.1016/0021-8693(87)90154-2
  • [36] 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 (89b:20087)
  • [37] G. Lusztig, Rationality properties of unipotent representations, J. Algebra 258 (2002), no. 1, 1-22. MR 1958895 (2004e:20017), https://doi.org/10.1016/S0021-8693(02)00514-8
  • [38] G. Lusztig, Hecke Algebras with Unequal Parameters, CRM Monograph Series, vol. 18, American Mathematical Society, Providence, RI, 2003. MR 1974442 (2004k:20011)
  • [39] 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
  • [40] J. Michel, The development version of the CHEVIE package of GAP3, preprint, available at arXiv:1310.7905.
  • [41] H. Naruse, $ W$-graphs for the irreducible representations of the Iwahori-Hecke algebras of type $ F_4$ and $ E_6$, private communication with M. Geck (January and July, 1998).
  • [42] Jian Yi Shi, The Kazhdan-Lusztig Cells in Certain Affine Weyl Groups, Lecture Notes in Mathematics, vol. 1179, Springer-Verlag, Berlin, 1986. MR 835214 (87i:20074)
  • [43] Jian-Yi Shi, Left cells containing a fully commutative element, J. Combin. Theory Ser. A 113 (2006), no. 3, 556-565. MR 2209710 (2007c:05200), https://doi.org/10.1016/j.jcta.2005.04.007
  • [44] Kazuhisa Takahashi, The left cells and their $ W$-graphs of Weyl group of type $ F_4$, Tokyo J. Math. 13 (1990), no. 2, 327-340. MR 1088235 (92a:20048), https://doi.org/10.3836/tjm/1270132265
  • [45] Chang Qing Tong, Left cells in Weyl group of type $ E_6$, Comm. Algebra 23 (1995), no. 13, 5031-5047. MR 1356119 (96k:20078), https://doi.org/10.1080/00927879508825517
  • [46] David A. Vogan Jr., A generalized $ \tau $-invariant for the primitive spectrum of a semisimple Lie algebra, Math. Ann. 242 (1979), no. 3, 209-224. MR 545215 (81e:17014), https://doi.org/10.1007/BF01420727
  • [47] David A. Vogan Jr., Ordering of the primitive spectrum of a semisimple Lie algebra, Math. Ann. 248 (1980), no. 3, 195-203. MR 575938 (81k:17006), https://doi.org/10.1007/BF01420525
  • [48] D. A. Vogan, The character table of $ E_8$, Notices of the Amer. Math. Soc. 9 (2007), 1022-1034; see also http://atlas.math.umd.edu/AIM_E8/technicaldetails.html.
  • [49] Nan Hua Xi, An approach to the connectedness of the left cells in affine Weyl groups, Bull. London Math. Soc. 21 (1989), no. 6, 557-561. MR 1018203 (90j:20110), https://doi.org/10.1112/blms/21.6.557

Similar Articles

Retrieve articles in Mathematics of Computation with MSC (2010): 20C40, 20C08, 20F55

Retrieve articles in all journals with MSC (2010): 20C40, 20C08, 20F55


Additional Information

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

Abbie Halls
Affiliation: 21 Rubislaw Terrace Lane, Aberdeen AB10 1XF, United Kingdom
Email: halls.abbie@gmail.com

DOI: https://doi.org/10.1090/mcom/2963
Received by editor(s): March 10, 2014
Received by editor(s) in revised form: April 14, 2014
Published electronically: May 8, 2015
Additional Notes: This work was supported by DFG Priority Programme SPP 1489
Article copyright: © Copyright 2015 American Mathematical Society

American Mathematical Society