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An inductive approach to Coxeter arrangements and Solomon’s descent algebra. (English) Zbl 1248.20006
Summary: In our recent paper [J. M. Douglass, G. Pfeiffer, G. Röhrle, Cohomology of Coxeter arrangements and Solomon’s descent algebra. arXiv:1101.2075v5], we claimed that both the group algebra of a finite Coxeter group \(W\) as well as the Orlik-Solomon algebra of \(W\) can be decomposed into a sum of induced one-dimensional representations of centralizers, one for each conjugacy class of elements of \(W\), and gave a uniform proof of this claim for symmetric groups. In this note, we outline an inductive approach to our conjecture. As an application of this method, we prove the inductive version of the conjecture for finite Coxeter groups of rank up to 2.

MSC:
20C08 Hecke algebras and their representations
05E10 Combinatorial aspects of representation theory
20F55 Reflection and Coxeter groups (group-theoretic aspects)
20C30 Representations of finite symmetric groups
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