Cohomology of Coxeter arrangements and Solomon’s descent algebra

Abstract

We refine a conjecture by Lehrer and Solomon on the structure of the Orlik-Solomon algebra of a finite Coxeter group W W and relate it to the descent algebra of W W . As a result, we claim that both the group algebra of W W and the Orlik-Solomon algebra of W W can be decomposed into a sum of induced one-dimensional representations of element centralizers, one for each conjugacy class of elements of W 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 ) (W, W_L) , where W W is arbitrary and W L W_L is a parabolic subgroup of W W , all of whose irreducible factors are of type A A .