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Cohomology of Coxeter arrangements and Solomon’s descent algebra. (English) Zbl 1343.20005
Summary: 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 \(A\).

MSC:
20C08 Hecke algebras and their representations
20F55 Reflection and Coxeter groups (group-theoretic aspects)
05E10 Combinatorial aspects of representation theory
20C30 Representations of finite symmetric groups
52C35 Arrangements of points, flats, hyperplanes (aspects of discrete geometry)
Software:
CHEVIE; GAP
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