The Representation Type of the Descent Algebras

TL;DR

This paper completes the classification of the representation type of descent algebras across various types, extending previous work to all types except E8, using theoretical and computational methods.

Contribution

It provides a comprehensive classification of descent algebra representation types for all types except E8, combining theoretical proofs and computer calculations.

Findings

Complete classification for all types except E8.

Type B classification achieved through purely theoretical methods.

Small cases in D and exceptional types analyzed with computer computations.

Abstract

Schocker classified the representation type of the descent algebra of type $\mathbb{A}$ over any field of characteristic zero. In an earlier paper, the authors extended this classification for type $\mathbb{A}$ to fields of positive characteristic. In this paper, we complete the classification for all other types except for $\mathbb{E}_8$. The proof for type $\mathbb{B}$ is entirely theoretical, while some small cases in type $\mathbb{D}$ and the exceptional types require computer computation to determine their Ext-quivers.