Quiver presentations for band algebras are defined over the integers

TL;DR

This paper establishes a uniform quiver presentation for band algebras over all fields by showing their integral semigroup algebra is isomorphic to a path algebra quotient, with applications to hyperplane face semigroup algebras.

Contribution

It proves that the integral semigroup algebra of a band is isomorphic to a quiver path algebra quotient, providing a unified framework for band algebras over any field.

Findings

Integral semigroup algebra of a band is isomorphic to a quiver path algebra quotient.

Provides a uniform bound quiver presentation for band algebras over all fields.

Answers a question on CW left regular bands related to hyperplane face semigroup algebras.

Abstract

A band is a semigroup in which each element is idempotent. In recent years, there has been a lot of activity on the representation theory of the subclass of left regular bands due to connections to Markov chains associated to hyperplane arrangements, oriented matroids, matroids and CAT(0) cube complexes. We prove here that the integral semigroup algebra of a band is isomorphic to the integral path algebra of a quiver modulo an admissible ideal. This leads to a uniform bound quiver presentation for band algebras over all fields. Also, we answer a question of Margolis, Saliola and Steinberg by proving that the integral semigroup algebra of a CW left regular band is isomorphic to the quotient of the integral path algebra of the Hasse diagram of its support semilattice modulo the ideal generated by the sum of all paths of length two. This includes, for example, hyperplane face semigroup algebras.