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

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.

To appear in Science China Mathematics