The Loewy length of the descent algebra of type D

TL;DR

This paper determines the Loewy length of the descent algebra of type D_{2m+1} as m+2, using geometric methods to analyze the algebra's quiver and reflection arrangement.

Contribution

It provides a precise calculation of the Loewy length for type D_{2m+1} descent algebra, confirming the bound with a geometric approach.

Findings

Loewy length of D_{2m+1} is m+2 for m ≥ 2

Longest path in the quiver has length at most m+1

Geometric approach links descent algebra to reflection arrangements

Abstract

The Loewy length of the descent algebra of type D_{2m+1} is shown to be m+2, for m \geq 2, by providing an upper bound that agrees with the lower bound in \cite{BonnafePfeiffer2006}. The bound is obtained by showing that the length of the longest path in the quiver of the descent algebra of D_{2m+1} is at most m+1. To achieve this bound, the geometric approach to the descent algebra is used, in which the descent algebra of a finite Coxeter group is identified with an algebra associated to the reflection arrangement of the group.