Quiver presentations for band algebras are defined over the integers
Benjamin Steinberg
TL;DR
The paper establishes a uniform integral quiver presentation for band algebras. For any connected band $B$, it constructs an explicit quiver $Q(B)$ and an admissible ideal $I$ with $\mathbb{Z}B \cong \mathbb{Z}Q(B)/I$, and, for every commutative ring $k$, proves $kB \cong kQ(B)/(k\otimes I)$, ensuring a field-independent presentation. It further develops a general bound-quiver framework over principal ideal domains and demonstrates that the same quiver description persists under base change. In the special case of CW left regular bands, the integral presentation reduces to the Hasse diagram of the support semilattice with $I$ generated by the sum of all length-2 paths, answering integrality questions about the bound-quiver construction and linking representation theory to topological and combinatorial structure.
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.