Combinatorics of descent algebras and graph coverings

TL;DR

This paper provides a combinatorial proof that the product of descent classes in symmetric groups results in a sum of descent classes, using graph coverings and extending to Coxeter groups.

Contribution

It introduces a new combinatorial proof technique based on graph coverings for descent algebra products, applicable to all Coxeter groups.

Findings

Product of descent classes is a sum of descent classes

Graph structure from weak order underpins the proof

Applicable to infinite Coxeter groups

Abstract

We give a direct combinatorial proof that the product of two descent classes in a symmetric group is a sum of descent classes. The proof is based on the fact that the group product gives a covering map when descent classes are endowed with the graph structure coming from the weak order. The main geometric argument is valid for any Coxeter group, even infinite ones for which the descent algebra does not exist.