Topology and monoid representations I: Foundations

TL;DR

This paper develops topological methods to compute Ext groups in monoid representation theory, determines the global dimension of affine transformation monoids, and characterizes homological epimorphisms using topology.

Contribution

It introduces a spectral sequence approach for Ext computations, provides a topological characterization of homological epimorphisms, and constructs projective resolutions for monoid modules.

Findings

Spectral sequence collapses on the E2-page over fields of good characteristic.

Global dimension of the affine transformation monoid algebra is determined.

Topological methods are used to construct projective resolutions.

Abstract

This paper aims to use topological methods to compute $\mathrm{Ext}$ 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 $E_2$-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 $G\wr S_n$).