Geodesic
An inductive approach to Coxeter arrangements and Solomon's descent algebra.
Journal of Algebraic Combinatorics, Tome 35 (2012) no. 2, pp. 215-235.

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Summary: In our recent paper (Douglass et al. arXiv:1101.2075 ( 2011)), 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.
Keywords: keywords Coxeter groups, reflection arrangements, descent algebra, dihedral groups 
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     title = {An inductive approach to {Coxeter} arrangements and {Solomon's} descent algebra.},
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Douglass, J.Matthew; Pfeiffer, Götz; Röhrle, Gerhard. An inductive approach to Coxeter arrangements and Solomon's descent algebra.. Journal of Algebraic Combinatorics, Tome 35 (2012) no. 2, pp. 215-235. http://geodesic.mathdoc.fr/item/JAC_2012__35_2_a5/