An Inductive Approach to Coxeter Arrangements and Solomon's Descent Algebra

TL;DR

This paper proposes an inductive method to decompose algebraic structures associated with finite Coxeter groups, providing proofs for groups of rank up to 2 and outlining a general approach for broader classes.

Contribution

It introduces an inductive approach to decompose Coxeter-related algebras, extending previous results and providing a new proof technique for low-rank cases.

Findings

Decomposition of group algebra and Orlik-Solomon algebra into induced representations.

Proof of the inductive conjecture for Coxeter groups of rank up to 2.

Outline of a general inductive approach for higher ranks.

Abstract

In a recent paper 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.