On the Quiver Presentation of the Descent Algebra of the Symmetric Group
Marcus Bishop, G\"otz Pfeiffer

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 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 is given by restricted partition refinement. Moreover, we describe certain families of relations and conjecture that for fixed , the finite set of relations from these families that are relevant for the descent algebra of generates the ideal of relations, and hence yields an explicit presentation by generators and relations of the algebra.
Peer Reviews
No public reviews on file for this paper yet. If you reviewed it on a platform where reviews are public (OpenReview, ICLR, NeurIPS, ICML), you can paste yours below so the community can read it here.
Videos
No videos yet. Explain this paper in a talk, walkthrough, or lecture? Add one.
