Simplicity of augmentation submodules for transformation monoids

TL;DR

This paper explores the conditions under which augmentation submodules are simple in finite transformation monoids, extending known results from permutation groups and providing numerous natural examples.

Contribution

It characterizes finite transformation monoids with simple augmentation submodules over various fields, generalizing from permutation groups and including diverse natural examples.

Findings

Characterization of monoids with simple augmentation submodules

Extension of group results to monoids for fields like , , and 

Examples include endomorphism monoids of complexes, posets, and graphs

Abstract

For finite permutation groups, simplicity of the augmentation submodule is equivalent to $2$-transitivity over the field of complex numbers. We note that this is not the case for transformation monoids. We characterize the finite transformation monoids whose augmentation submodules are simple for a field $\mathbb{F}$ (assuming the answer is known for groups, which is the case for $\mathbb C$, $\mathbb R$, and $\mathbb Q$) and provide many interesting and natural examples such as endomorphism monoids of connected simplicial complexes, posets, and graphs (the latter with simplicial mappings).