On the quiver presentation of the descent algebra of the symmetric group

We describe a presentation of the descent algebra of the symmetric group Sn 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 short new proof of the known fact that the quiver of the descent algebra of Sn is given by restricted partition refinement. Moreover, we describe certain families of relations and conjecture that for fixed n∈N the finite set of relations from these families that are relevant for the descent algebra of Sn generates the ideal of relations of an explicit quiver presentation of that algebra.