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
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by J. Matthew Douglass, Götz Pfeiffer and Gerhard Röhrle PDF

Trans. Amer. Math. Soc. 366 (2014), 5379-5407 Request permission

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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