Cohomology of Coxeter arrangements and Solomon's descent algebra

2011/01/11 by Douglass, J. Matthew, Pfeiffer, Goetz, Roehrle, Gerhard
#05E10 #20F55 #52C35 #Combinatorics (math.CO) #FOS: Mathematics #Group Theory (math.GR) #Representation Theory (math.RT)

paper · doi:10.48550/arxiv.1101.2075

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, as well as 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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