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

TL;DR

This paper extends a quiver-based presentation method for the descent algebra from symmetric groups to hyperoctahedral groups, offering a new proof and a practical way to compute relations.

Contribution

It generalizes a previous construction to the hyperoctahedral group, providing a simplified proof and a method for calculating the algebra's relations.

Findings

Provides a simple proof of the quiver of the hyperoctahedral descent algebra

Develops a straightforward method for calculating relations in the algebra

Extends the quiver presentation technique to a new class of groups

Abstract

In a recent article we introduced a mechanism for producing a presentation of the descent algebra of the symmetric group as a quiver with relations, the mechanism arising from a new construction of the descent algebra as a homomorphic image of an algebra of binary forests. Here we extend the method to construct a similar presentation of the descent algebra of the hyperoctahedral group, providing a simple proof of the known formula for the quiver of this algebra and a straightforward method for calculating the relations.