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

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.