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On the quiver presentation of the descent algebra of the symmetric group. (English) Zbl 1291.20003
Summary: We describe a presentation of the descent algebra of the symmetric group $$\mathfrak S_n$$ 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 short new proof of the known fact that the quiver of the descent algebra of $$\mathfrak S_n$$ 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 $$\mathfrak S_n$$ generates the ideal of relations of an explicit quiver presentation of that algebra.

##### MSC:
 20C08 Hecke algebras and their representations 16G20 Representations of quivers and partially ordered sets 20C30 Representations of finite symmetric groups 17B01 Identities, free Lie (super)algebras 20F55 Reflection and Coxeter groups (group-theoretic aspects) 05E10 Combinatorial aspects of representation theory
##### Keywords:
descent algebras; symmetric groups; quivers; presentations
GAP
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##### References:
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