Simplicity of Augmentation Submodules for Transformation Monoids

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 𝔽 (assuming the answer is known for groups, which is the case for ℂ, ℝ, and ℚ) and provide many interesting and natural examples such as endomorphism monoids of connected simplicial complexes, posets, and graphs (the latter with simplicial mappings).