On the quiver of the descent algebra

TL;DR

This paper investigates the structure of the descent algebra's quiver for finite Coxeter groups, deriving specific cases for types A and B by analyzing the algebra's relation to reflection arrangements.

Contribution

It provides a novel approach to understanding the descent algebra's quiver through reflection arrangements, including explicit derivations for types A and B.

Findings

Derived the quiver of the descent algebra for types A and B

Connected the descent algebra structure to reflection arrangements

Enhanced understanding of the algebra's combinatorial properties

Abstract

We study the quiver of the descent algebra of a finite Coxeter group W. The results include a derivation of the quiver of the descent algebra of types A and B. Our approach is to study the descent algebra as an algebra constructed from the reflection arrangement associated to W.