Topology and monoid representations I: Foundations

2023/06/28 by Benjamin Steinberg, Steinberg, Benjamin · 1 citation
Mathematics · #Homotopy and Cohomology in Algebraic Topology #Advanced Topics in Algebra #Rings, Modules, and Algebras

paper · pdf · doi:10.48550/arxiv.2306.16379

Abstract

This paper aims to use topological methods to compute Ext between an irreducible representation of a finite monoid inflated from its group completion and one inflated from its group of units, or more generally coinduced from a maximal subgroup, via a spectral sequence that collapses on the E₂-page over fields of good characteristic. As an application, we determine the global dimension of the algebra of the monoid of all affine transformations of a vector space over a finite field. We provide a topological characterization of when a monoid homomorphism induces a homological epimorphism of monoid algebras and apply it to semidirect products. Topology is used to construct projective resolutions of modules inflated from the group completion for sufficiently nice monoids. A sequel paper will use these results to study the representation theory Hsiao's monoid of ordered G-partitions (connected to the Mantaci-Reutenauer descent algebra for the wreath product G\wr S_n).

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• Left regular bands with symmetry

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