Quiver mutations and Boolean reflection monoids

TL;DR

This paper explores the connection between quiver mutations and Boolean reflection monoids, providing new presentations, showing their compatibility with mutations, and establishing their cellular algebra structure.

Contribution

It introduces a family of presentations for Boolean reflection monoids compatible with quiver mutations and links these to automorphisms and cellular algebra properties.

Findings

Presentations of Boolean reflection monoids are compatible with quiver mutations.

Inner automorphisms can be constructed via mutation sequences.

Semigroup algebras of Boolean reflection monoids are cellular.

Abstract

In 2010, Everitt and Fountain introduced the concept of reflection monoids. The Boolean reflection monoids form a family of reflection monoids (symmetric inverse semigroups are Boolean reflection monoids of type $A$). In this paper, we give a family of presentations of Boolean reflection monoids and show how these presentations are compatible with quiver mutations of orientations of Dynkin diagrams with frozen vertices. Our results recover the presentations of Boolean reflection monoids given by Everitt and Fountain and the presentations of symmetric inverse semigroups given by Popova respectively. Surprisingly, inner by diagram automorphisms of irreducible Weyl groups and Boolean reflection monoids can be constructed by sequences of mutations preserving the same underlying diagrams. Besides, we show that semigroup algebras of Boolean reflection monoids are cellular algebras.