Quivers of monoids with basic algebras

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.Comment: Minor corrections and improvements to exposition were made. Some theorem statements were simplified. Also we made a language change. Several of our results are more naturally expressed using the language of Karoubi envelopes and irreducible morphisms. There are no substantial changes in actual result