Marcus Bishop, and Götz Pfeiffer,
We describe a presentation for the descent algebra of the symmetric group
S_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 ofS_n is given by restricted partition refinement. Moreover, we describe certain families of relations and conjecture that for fixedn inN , the finite set of relations from these families that are relevant for the descent algebra ofS_n generates the ideal of relations, and hence yields an explicit presentation by generators and relations of the algebra.
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