Quiver Presentations for Descent Algebras of Exceptional Type

TL;DR

This paper develops a combinatorial framework to compute quiver presentations of descent algebras for finite Coxeter groups, and applies it to exceptional types such as E6, E7, E8, F4, H3, H4, and I2(m).

Contribution

It introduces a systematic method for deriving quiver presentations of descent algebras for complex Coxeter groups, extending previous work to exceptional types.

Findings

Quiver presentations for descent algebras of exceptional Coxeter groups are explicitly determined.

The framework simplifies the computation process for complex group types.

Results facilitate further algebraic and representation-theoretic studies.

Abstract

The descent algebra of a finite Coxeter group $W$ is a basic algebra, and as such it has a presentation as quiver with relations. In recent work, we have developed a combinatorial framework which allows us to systematically compute such a quiver presentation for a Coxeter group of a given type. In this article, we use that framework to determine quiver presentations for the descent algebras of the Coxeter groups of exceptional or non-crystallographic type, i.e., of type $E_6$, $E_7$, $E_8$, $F_4$, $H_3$, $H_4$ or $I_2(m)$.