Remote Access Transactions of the American Mathematical Society
Green Open Access

Transactions of the American Mathematical Society

ISSN 1088-6850(online) ISSN 0002-9947(print)

 
 

 

Cohomology of Coxeter arrangements and Solomon's descent algebra


Authors: J. Matthew Douglass, Götz Pfeiffer and Gerhard Röhrle
Journal: Trans. Amer. Math. Soc. 366 (2014), 5379-5407
MSC (2010): Primary 20F55; Secondary 05E10, 52C35
DOI: https://doi.org/10.1090/S0002-9947-2014-06060-1
Published electronically: June 19, 2014
MathSciNet review: 3240927
Full-text PDF

Abstract | References | Similar Articles | Additional Information

Abstract: We refine a conjecture by Lehrer and Solomon on the structure of the Orlik-Solomon algebra of a finite Coxeter group $ W$ and relate it to the descent algebra of $ W$. As a result, we claim that both the group algebra of $ W$ and the Orlik-Solomon algebra of $ W$ can be decomposed into a sum of induced one-dimensional representations of element centralizers, one for each conjugacy class of elements of $ W$. We give a uniform proof of the claim for symmetric groups. In addition, we prove that a relative version of the conjecture holds for every pair $ (W, W_L)$, where $ W$ is arbitrary and $ W_L$ is a parabolic subgroup of $ W$, all of whose irreducible factors are of type $ A$.


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

  • [1] Hélène Barcelo and Nantel Bergeron, The Orlik-Solomon algebra on the partition lattice and the free Lie algebra, J. Combin. Theory Ser. A 55 (1990), no. 1, 80-92. MR 1070017 (91i:05119), https://doi.org/10.1016/0097-3165(90)90049-3
  • [2] François Bergeron and Nantel Bergeron, Orthogonal idempotents in the descent algebra of $ B_n$ and applications, J. Pure Appl. Algebra 79 (1992), no. 2, 109-129. MR 1163285 (93f:20054), https://doi.org/10.1016/0022-4049(92)90153-7
  • [3] F. Bergeron, N. Bergeron, and A. M. Garsia,
    Idempotents for the free Lie algebra and $ q$-enumeration.
    In Invariant theory and tableaux (Minneapolis, MN, 1988), volume 19 of IMA Vol. Math. Appl., pages 166-190. Springer, New York, 1990.
  • [4] F. Bergeron, N. Bergeron, R. B. Howlett, and D. E. Taylor, A decomposition of the descent algebra of a finite Coxeter group, J. Algebraic Combin. 1 (1992), no. 1, 23-44. MR 1162640 (93g:20079), https://doi.org/10.1023/A:1022481230120
  • [5] Nantel Bergeron, Hyperoctahedral operations on Hochschild homology, Adv. Math. 110 (1995), no. 2, 255-276. MR 1317618 (96h:16009), https://doi.org/10.1006/aima.1995.1011
  • [6] Marcus Bishop, J. Matthew Douglass, Götz Pfeiffer, and Gerhard Röhrle, Computations for Coxeter arrangements and Solomon's descent algebra: Groups of rank three and four, J. Symbolic Comput. 50 (2013), 139-158. MR 2996872, https://doi.org/10.1016/j.jsc.2012.06.001
  • [7] M. Bishop, J.M. Douglass, G. Pfeiffer, and G. Röhrle. Computations for Coxeter arrangements and Solomon's descent algebra II: Groups of rank five and six, J. Algebra 377 (2013), 320-332. MR 3008911
  • [8] M. Bishop, J.M. Douglass, G. Pfeiffer, and G. Röhrle, Computations for Coxeter arrangements and Solomon's descent algebra III: Groups of rank seven and eight, Arxiv: http://arxiv.org/abs/1403.6227
  • [9] J. Blair and G. I. Lehrer, Cohomology actions and centralisers in unitary reflection groups, Proc. London Math. Soc. (3) 83 (2001), no. 3, 582-604. MR 1851083 (2002g:14093), https://doi.org/10.1112/plms/83.3.582
  • [10] Angeline Brandt, The free Lie ring and Lie representations of the full linear group, Trans. Amer. Math. Soc. 56 (1944), 528-536. MR 0011305 (6,146d)
  • [11] Egbert Brieskorn, Sur les groupes de tresses [d'après V. I. Arnold], Séminaire Bourbaki, 24ème année (1971/1972), Exp. No. 401, Springer, Berlin, 1973, pp. 21-44. Lecture Notes in Math., Vol. 317 (French). MR 0422674 (54 #10660)
  • [12] J. Matthew Douglass, On the cohomology of an arrangement of type $ B_l$, J. Algebra 147 (1992), no. 2, 265-282. MR 1161294 (93d:52017), https://doi.org/10.1016/0021-8693(92)90206-2
  • [13] J. Matthew Douglass, Götz Pfeiffer, and Gerhard Röhrle, An inductive approach to Coxeter arrangements and Solomon's descent algebra, J. Algebraic Combin. 35 (2012), no. 2, 215-235. MR 2886288 (2012m:05431), https://doi.org/10.1007/s10801-011-0301-9
  • [14] G. Felder and A. P. Veselov, Coxeter group actions on the complement of hyperplanes and special involutions, J. Eur. Math. Soc. (JEMS) 7 (2005), no. 1, 101-116. MR 2120992 (2006c:20082), https://doi.org/10.4171/JEMS/23
  • [15] A. M. Garsia and C. Reutenauer, A decomposition of Solomon's descent algebra, Adv. Math. 77 (1989), no. 2, 189-262. MR 1020585 (91c:20007), https://doi.org/10.1016/0001-8708(89)90020-0
  • [16] 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 (99m:20017), https://doi.org/10.1007/BF01190329
  • [17] M. Geck and G. Malle, Frobenius-Schur indicators of unipotent characters and the twisted involution module, Represent. Theory 17 (2013), 180-198. MR 3037782
  • [18] 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)
  • [19] Phil Hanlon, The action of $ S_n$ on the components of the Hodge decomposition of Hochschild homology, Michigan Math. J. 37 (1990), no. 1, 105-124. MR 1042517 (91g:20013), https://doi.org/10.1307/mmj/1029004069
  • [20] Robert B. Howlett, Normalizers of parabolic subgroups of reflection groups, J. London Math. Soc. (2) 21 (1980), no. 1, 62-80. MR 576184 (81g:20094), https://doi.org/10.1112/jlms/s2-21.1.62
  • [21] R. B. Howlett and G. I. Lehrer, Duality in the normalizer of a parabolic subgroup of a finite Coxeter group, Bull. London Math. Soc. 14 (1982), no. 2, 133-136. MR 647196 (83e:20049), https://doi.org/10.1112/blms/14.2.133
  • [22] A. A. Klyachko, Lie elements in the tensor algebra, Siberian Math. J. 15 (1974), 914-921.
  • [23] Matjaž Konvalinka, Götz Pfeiffer, and Claas E. Röver, A note on element centralizers in finite Coxeter groups, J. Group Theory 14 (2011), no. 5, 727-745. MR 2831968 (2012j:20125), https://doi.org/10.1515/JGT.2010.074
  • [24] 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
  • [25] G. I. Lehrer, On the Poincaré series associated with Coxeter group actions on complements of hyperplanes, J. London Math. Soc. (2) 36 (1987), no. 2, 275-294. MR 906148 (88k:32037), https://doi.org/10.1112/jlms/s2-36.2.275
  • [26] G. I. Lehrer and Louis Solomon, On the action of the symmetric group on the cohomology of the complement of its reflecting hyperplanes, J. Algebra 104 (1986), no. 2, 410-424. MR 866785 (88a:32017), https://doi.org/10.1016/0021-8693(86)90225-5
  • [27] G. Lusztig and D. Vogan, Hecke algebras and involutions in Weyl groups, Bull. Inst. Math. Acad. Sin. (N.S.) 7 (2012), no. 3, 323-354. MR 3051317
  • [28] Peter Orlik and Louis Solomon, Combinatorics and topology of complements of hyperplanes, Invent. Math. 56 (1980), no. 2, 167-189. MR 558866 (81e:32015), https://doi.org/10.1007/BF01392549
  • [29] Peter Orlik and Louis Solomon, Coxeter arrangements, Singularities, Part 2 (Arcata, Calif., 1981) Proc. Sympos. Pure Math., vol. 40, Amer. Math. Soc., Providence, RI, 1983, pp. 269-291. MR 713255 (85b:32016)
  • [30] Peter Orlik and Hiroaki Terao, Arrangements of hyperplanes, Grundlehren der Mathematischen Wissenschaften [Fundamental Principles of Mathematical Sciences], vol. 300, Springer-Verlag, Berlin, 1992. MR 1217488 (94e:52014)
  • [31] Christophe Reutenauer, Free Lie algebras, London Mathematical Society Monographs. New Series, vol. 7, The Clarendon Press Oxford University Press, New York, 1993. Oxford Science Publications. MR 1231799 (94j:17002)
  • [32] M. Schonert et al.
    GAP - Groups, Algorithms, and Programming - version 3 release 4,
    Lehrstuhl D fur Mathematik, Rheinisch Westfalische Technische Hochschule, Aachen, Germany, 1997.
  • [33] Manfred Schocker, Über die höheren Lie-Darstellungen der symmetrischen Gruppen, Bayreuth. Math. Schr. 63 (2001), 103-263 (German, with English summary). MR 1867282 (2002i:20018)
  • [34] Louis Solomon, A decomposition of the group algebra of a finite Coxeter group, J. Algebra 9 (1968), 220-239. MR 0232868 (38 #1191)
  • [35] Louis Solomon, A Mackey formula in the group ring of a Coxeter group, J. Algebra 41 (1976), no. 2, 255-264. MR 0444756 (56 #3104)
  • [36] Richard P. Stanley, Some aspects of groups acting on finite posets, J. Combin. Theory Ser. A 32 (1982), no. 2, 132-161. MR 654618 (83d:06002), https://doi.org/10.1016/0097-3165(82)90017-6
  • [37] Robert Steinberg, Differential equations invariant under finite reflection groups, Trans. Amer. Math. Soc. 112 (1964), 392-400. MR 0167535 (29 #4807)
  • [38] Jacques Thévenaz, $ G$-algebras and modular representation theory, Oxford Mathematical Monographs, The Clarendon Press Oxford University Press, New York, 1995. Oxford Science Publications. MR 1365077 (96j:20017)
  • [39] Franz Wever, Über Invarianten in Lie'schen Ringen, Math. Ann. 120 (1949), 563-580 (German). MR 0029893 (10,676e)

Similar Articles

Retrieve articles in Transactions of the American Mathematical Society with MSC (2010): 20F55, 05E10, 52C35

Retrieve articles in all journals with MSC (2010): 20F55, 05E10, 52C35


Additional Information

J. Matthew Douglass
Affiliation: Department of Mathematics, University of North Texas, Denton, Texas 76203
Email: douglass@unt.edu

Götz Pfeiffer
Affiliation: School of Mathematics, Statistics and Applied Mathematics, National University of Ireland Galway, Galway, Ireland
Email: goetz.pfeiffer@nuigalway.ie

Gerhard Röhrle
Affiliation: Fakultät für Mathematik, Ruhr-Universität Bochum, D-44780 Bochum, Germany
Email: gerhard.roehrle@rub.de

DOI: https://doi.org/10.1090/S0002-9947-2014-06060-1
Received by editor(s): July 16, 2012
Received by editor(s) in revised form: December 4, 2012
Published electronically: June 19, 2014
Additional Notes: The authors would like to thank their charming wives for their unwavering support during the preparation of this paper.
Article copyright: © Copyright 2014 American Mathematical Society
The copyright for this article reverts to public domain 28 years after publication.

American Mathematical Society