Bijection between Conjugacy Classes and Irreducible Representations of Finite Inverse Semigroups

TL;DR

This paper establishes a bijection between conjugacy classes and irreducible representations of finite inverse semigroups over certain fields, extending understanding of their algebraic structure.

Contribution

It proves a bijection between conjugacy classes and irreducible representations for finite inverse semigroups under specific field characteristic conditions.

Findings

Bijection holds when the characteristic of the field is zero or prime not dividing maximal subgroup orders.

Extends classical results from group theory to inverse semigroups.

Provides a framework for analyzing representations of finite inverse semigroups.

Abstract

In this paper we show that the irreducible representations of a finite inverse semigroup $S$ over an algebraically closed field $F$ are in bijection with the conjugacy classes of $S$ if the characteristic of $F$ is zero or a prime number that does not divide the order of any maximal subgroup of $S$.