On the Quiver Presentation of the Descent Algebra of the Symmetric Group

TL;DR

This paper presents a new quiver with relations for the descent algebra of the symmetric group, using a forest of binary trees approach, and conjectures a finite generating set for the relations.

Contribution

It introduces a novel construction of the descent algebra as a homomorphic image of forest algebras, providing a new proof and a conjecture on finite relations.

Findings

New quiver presentation for the descent algebra.

A short proof that the quiver is given by restricted partition refinement.

Conjecture that a finite set of relations generates the ideal of relations.

Abstract

We describe a presentation for the descent algebra of the symmetric group $\sym{n}$ as a quiver with relations. This presentation arises from a new construction of the descent algebra as a homomorphic image of an algebra of forests of binary trees which can be identified with a subspace of the free Lie algebra. In this setting, we provide a new short proof of the known fact that the quiver of the descent algebra of $\sym{n}$ is given by restricted partition refinement. Moreover, we describe certain families of relations and conjecture that for fixed $n\in\mathbb{N}$, the finite set of relations from these families that are relevant for the descent algebra of $\sym{n}$ generates the ideal of relations, and hence yields an explicit presentation by generators and relations of the algebra.