Simplicity of augmentation submodules for transformation monoids

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).