Quivers of monoids with basic algebras

TL;DR

This paper computes the quiver of monoids with basic algebras over algebraically closed fields of characteristic zero, reducing the problem to group algebra computations for a broad class called rectangular monoids.

Contribution

It introduces a reduction method for computing quivers of rectangular monoids, linking semigroup theory with representation theory, and provides explicit descriptions for R-trivial monoids.

Findings

Complete quiver computation for monoids with basic algebras in good characteristic.

Reduction of quiver computation to group algebra problems for rectangular monoids.

Semigroup theoretic description of projective indecomposables and Cartan matrix for R-trivial monoids.

Abstract

We compute the quiver of any monoid that has a basic algebra over an algebraically closed field of characteristic zero. More generally, we reduce the computation of the quiver over a splitting field of a class of monoids that we term rectangular monoids (in the semigroup theory literature the class is known as $\mathbf{DO}$) to representation theoretic computations for group algebras of maximal subgroups. Hence in good characteristic for the maximal subgroups, this gives an essentially complete computation. Since groups are examples of rectangular monoids, we cannot hope to do better than this. For the subclass of $\mathscr R$-trivial monoids, we also provide a semigroup theoretic description of the projective indecomposables and compute the Cartan matrix.