Cohomology of coxeter arrangements and solomon’s descent algebra
Douglass, J. Matthew
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Douglass, J. Matthew; Pfeiffer, Götz; Röhrle, Gerhard (2014). Cohomology of coxeter arrangements and solomon’s descent algebra. Transactions of the American Mathematical Society 366 (10), 5379-5407
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.