Cohomology of Coxeter arrangements and Solomon’s descent algebra

Douglass, J.; Pfeiffer, Götz; Röhrle, Gerhard

doi:10.1090/S0002-9947-2014-06060-1

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 and relate it to the descent algebra of . As a result, we claim that both the group algebra of and the Orlik-Solomon algebra of can be decomposed into a sum of induced one-dimensional representations of element centralizers, one for each conjugacy class of elements of . 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 , where is arbitrary and is a parabolic subgroup of , all of whose irreducible factors are of type .

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