A Quiver Presentation for Solomon's Descent Algebra

TL;DR

This paper constructs a quiver presentation for Solomon's descent algebra of finite Coxeter groups, providing a new algebraic framework and an algorithm for explicit computation of its quiver structure.

Contribution

It introduces a novel construction of the descent algebra as a quotient of a subalgebra of a path algebra, enabling explicit quiver presentations and computational methods.

Findings

Provides a general construction for the quiver of the descent algebra.

Develops an algorithm to compute the quiver presentation for any finite Coxeter group.

Offers insights into the structure of the descent algebra via quiver theory.

Abstract

The descent algebra $\Sigma(W)$ is a subalgebra of the group algebra $\Q W$ of a finite Coxeter group $W$, which supports a homomorphism with nilpotent kernel and commutative image in the character ring of $W$. Thus $\Sigma(W)$ is a basic algebra, and as such it has a presentation as a quiver with relations. Here we construct $\Sigma(W)$ as a quotient of a subalgebra of the path algebra of the Hasse diagram of the Boolean lattice of all subsets of $S$, the set of simple reflections in $W$. From this construction we obtain some general information about the quiver of $\Sigma(W)$ and an algorithm for the construction of a quiver presentation for the descent algebra $\Sigma(W)$ of any given finite Coxeter group $W$.