Cohomology of Coxeter arrangements and Solomon's descent algebra

TL;DR

This paper refines a conjecture linking the cohomology of Coxeter arrangements with Solomon's descent algebra, showing decompositions of group and algebra structures into induced representations, with proofs for symmetric groups and certain parabolic subgroups.

Contribution

It provides a uniform proof for symmetric groups and extends the conjecture to specific parabolic subgroups, deepening understanding of Coxeter group algebraic structures.

Findings

Decomposition of the group algebra of W into induced representations.

Decomposition of the Orlik-Solomon algebra into induced representations.

Validation of the conjecture for symmetric groups and certain parabolic subgroups.

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 $A$.