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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
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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$.

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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.

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