Cohomology of Coxeter arrangements and Solomon's descent algebra

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.The authors would like to acknowledge support from the DFG-priority programme SPP1489 “Algorithmic and Experimental Methods in Algebra, Geometry, and Number Theory”. Part of the research for this paper was carried out while the authors were staying at the Mathematical Research Institute Oberwolfach supported by the “Research in Pairs” programme. The second author wishes to acknowledge support from Science Foundation Ireland.peer-reviewe