The Representation Type of the Descent Algebras of Type $\mathbb{A}$

TL;DR

This paper classifies when descent algebras of type A have finite or wild representation types, showing they are mostly wild except for small degrees, using algebra homomorphisms and representation theory techniques.

Contribution

It provides a complete classification of the representation type of type A descent algebras in positive characteristic, highlighting the finite cases and the general wild behavior.

Findings

Finite representation type only for small degrees

Most descent algebras are wild in larger degrees

Uses algebra homomorphisms and finite-dimensional algebra techniques

Abstract

We classify the representation type of the descent algebras of type $\A$ in the positive characteristic case. The algebras have finite representation type only for a few small degrees; otherwise, they are wild. Our main reduction method relies on a surjective algebra homomorphism from a descent algebra of type $\A$ to another of lower degree. For small degree cases, we employ techniques from the representation theory of finite-dimensional algebras.