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Simplicity of Augmentation Submodules for Transformation Monoids

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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 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).

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Acknowledgments

The first author was partially supported by CMUP, which is financed by national funds through FCT - Fundação para a Ciência e a Tecnologia, I.P., under the project with reference UIDB/00144/2020. He also acknowledges FCT for a contract based on the “Lei do Emprego Cientfico” (DL 57/2016). The second author was supported by NSA MSP #H98230-16-1-0047 and PSC-CUNY. This work was performed while the first author was visiting the City College of New York. He thanks the College for its warm hospitality.

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  1. Centro de Matemática e Departamento de Matemática, Faculdade de Ciências, Universidade do Porto, Rua do Campo Alegre, 687, 4169-007, Porto, Portugal

    M. H. Shahzamanian

  2. Department of Mathematics, City College of New York, Convent Avenue at 138th Street, New York, NY, 10031, USA

    B. Steinberg

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  1. M. H. Shahzamanian
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Shahzamanian, M.H., Steinberg, B. Simplicity of Augmentation Submodules for Transformation Monoids. Algebr Represent Theor 24, 1029–1051 (2021). https://doi.org/10.1007/s10468-020-09977-7

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