Computation of Kazhdan-Lusztig polynomials and some applications to finite groups

Lübeck, Frank

doi:10.1090/tran/8037

Computation of Kazhdan-Lusztig polynomials and some applications to finite groups
HTML articles powered by AMS MathViewer

by Frank Lübeck PDF

Trans. Amer. Math. Soc. 373 (2020), 2331-2347 Request permission

Abstract:

We discuss a practical algorithm to compute parabolic Kazhdan-Lusztig polynomials. As an application we compute Kazhdan-Lusztig polynomials which are needed to evaluate a character formula for reductive groups due to Lusztig.

Some coefficients of these polynomials have interesting interpretations for certain finite groups. We find examples of finite dimensional modules for finite groups with much higher dimensional first cohomology group than in all previously known cases.

Some of these examples lead to the construction of finite groups with many maximal subgroups, contradicting an old conjecture by G. E. Wall.

References

Henning Haahr Andersen, An inversion formula for the Kazhdan-Lusztig polynomials for affine Weyl groups, Adv. in Math. 60 (1986), no. 2, 125–153. MR 840301, DOI 10.1016/S0001-8708(86)80008-1
H. H. Andersen, J. C. Jantzen, and W. Soergel, Representations of quantum groups at a $p$th root of unity and of semisimple groups in characteristic $p$: independence of $p$, Astérisque 220 (1994), 321 (English, with English and French summaries). MR 1272539
Christopher P. Bendel, Brian D. Boe, Christopher M. Drupieski, Daniel K. Nakano, Brian J. Parshall, Cornelius Pillen, and Caroline B. Wright, Bounding the dimensions of rational cohomology groups, Developments and retrospectives in Lie theory, Dev. Math., vol. 38, Springer, Cham, 2014, pp. 51–69. MR 3308777, DOI 10.1007/978-3-319-09804-3_{2}
Christopher P. Bendel, Daniel K. Nakano, and Cornelius Pillen, On comparing the cohomology of algebraic groups, finite Chevalley groups and Frobenius kernels, J. Pure Appl. Algebra 163 (2001), no. 2, 119–146. MR 1846657, DOI 10.1016/S0022-4049(01)00024-X
D. J. Benson, Representations and cohomology. I, 2nd ed., Cambridge Studies in Advanced Mathematics, vol. 30, Cambridge University Press, Cambridge, 1998. Basic representation theory of finite groups and associative algebras. MR 1644252
B. Boe, The maximum length in the restricted region, (unpublished), 1998.
John N. Bray and Robert A. Wilson, Examples of 3-dimensional 1-cohomology for absolutely irreducible modules of finite simple groups, J. Group Theory 11 (2008), no. 5, 669–673. MR 2446148, DOI 10.1515/JGT.2008.042
Roger W. Carter, Finite groups of Lie type, Wiley Classics Library, John Wiley & Sons, Ltd., Chichester, 1993. Conjugacy classes and complex characters; Reprint of the 1985 original; A Wiley-Interscience Publication. MR 1266626
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, DOI 10.1016/0021-8693(87)90232-8
Fokko du Cloux, Computing Kazhdan-Lusztig polynomials for arbitrary Coxeter groups, Experiment. Math. 11 (2002), no. 3, 371–381. MR 1959749, DOI 10.1080/10586458.2002.10504482
Fokko du Cloux, The Coxeter 3 program, http://math.univ-lyon1.fr/~ducloux/coxeter/coxeter3/, 2005.
Ben Elias and Geordie Williamson, The Hodge theory of Soergel bimodules, Ann. of Math. (2) 180 (2014), no. 3, 1089–1136. MR 3245013, DOI 10.4007/annals.2014.180.3.6
Peter Fiebig, An upper bound on the exceptional characteristics for Lusztig’s character formula, J. Reine Angew. Math. 673 (2012), 1–31. MR 2999126, DOI 10.1515/CRELLE.2011.170
GAP – Groups, Algorithms, and Programming, Version 4.7.6, http://www.gap-system.org, Nov 2014.
Meinolf Geck, Gerhard Hiss, Frank Lübeck, Gunter Malle, and Götz Pfeiffer, CHEVIE—a system for computing and processing generic character tables, Appl. Algebra Engrg. Comm. Comput. 7 (1996), no. 3, 175–210. Computational methods in Lie theory (Essen, 1994). MR 1486215, DOI 10.1007/BF01190329
Robert M. Guralnick, The dimension of the first cohomology group, Representation theory, II (Ottawa, Ont., 1984) Lecture Notes in Math., vol. 1178, Springer, Berlin, 1986, pp. 94–97. MR 842479, DOI 10.1007/BFb0075290
Robert M. Guralnick and Corneliu Hoffman, The first cohomology group and generation of simple groups, Groups and geometries (Siena, 1996) Trends Math., Birkhäuser, Basel, 1998, pp. 81–89. MR 1644977, DOI 10.1007/bf01214004
Robert M. Guralnick and Pham Huu Tiep, First cohomology groups of Chevalley groups in cross characteristic, Ann. of Math. (2) 174 (2011), no. 1, 543–559. MR 2811608, DOI 10.4007/annals.2011.174.1.16
Marcel Herzog and Olaf Manz, On the number of subgroups in finite solvable groups, J. Austral. Math. Soc. Ser. A 58 (1995), no. 1, 134–141. MR 1313892, DOI 10.1017/S1446788700038155
James E. Humphreys, Reflection groups and Coxeter groups, Cambridge Studies in Advanced Mathematics, vol. 29, Cambridge University Press, Cambridge, 1990. MR 1066460, DOI 10.1017/CBO9780511623646
Bertram Huppert and Norman Blackburn, Finite groups. II, Grundlehren der Mathematischen Wissenschaften [Fundamental Principles of Mathematical Sciences], vol. 242, Springer-Verlag, Berlin-New York, 1982. AMD, 44. MR 650245
Jens Carsten Jantzen, Representations of algebraic groups, 2nd ed., Mathematical Surveys and Monographs, vol. 107, American Mathematical Society, Providence, RI, 2003. MR 2015057
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
Martin W. Liebeck, Laszlo Pyber, and Aner Shalev, On a conjecture of G. E. Wall, J. Algebra 317 (2007), no. 1, 184–197. MR 2360145, DOI 10.1016/j.jalgebra.2006.10.047
George Lusztig, Some problems in the representation theory of finite Chevalley groups, The Santa Cruz Conference on Finite Groups (Univ. California, Santa Cruz, Calif., 1979) Proc. Sympos. Pure Math., vol. 37, Amer. Math. Soc., Providence, R.I., 1980, pp. 313–317. MR 604598
Benjamin Newton, On the number of maximal subgroups of a finite solvable group, Arch. Math. (Basel) 96 (2011), no. 6, 501–506. MR 2821467, DOI 10.1007/s00013-011-0252-3
Alison E. Parker and David I. Stewart, First cohomology groups for finite groups of Lie type in defining characteristic, Bull. Lond. Math. Soc. 46 (2014), no. 2, 227–238. MR 3194742, DOI 10.1112/blms/bdt087
Leonard L. Scott, Some new examples in 1-cohomology, J. Algebra 260 (2003), no. 1, 416–425. Special issue celebrating the 80th birthday of Robert Steinberg. MR 1976701, DOI 10.1016/S0021-8693(02)00667-1
Leonard L. Scott and Timothy Sprowl, Computing individual Kazhdan-Lusztig basis elements, J. Symbolic Comput. 73 (2016), 244–249. MR 3385961, DOI 10.1016/j.jsc.2015.05.003
Wolfgang Soergel, Kazhdan-Lusztig-Polynome und eine Kombinatorik für Kipp-Moduln, Represent. Theory 1 (1997), 37–68 (German, with English summary). MR 1445511, DOI 10.1090/S1088-4165-97-00006-X
Robert Steinberg, Representations of algebraic groups, Nagoya Math. J. 22 (1963), 33–56. MR 155937, DOI 10.1017/S0027763000011016
G. E. Wall, Some applications of the Eulerian functions of a finite group, J. Austral. Math. Soc. 2 (1961/1962), 35–59. MR 0125156, DOI 10.1017/S1446788700026367
Geordie Williamson, Schubert calculus and torsion explosion, J. Amer. Math. Soc. 30 (2017), no. 4, 1023–1046. With a joint appendix with Alex Kontorovich and Peter J. McNamara. MR 3671935, DOI 10.1090/jams/868