Linear conjugacy

TL;DR

This paper characterizes when two elements of finite semigroups and groups are conjugate under all linear representations over any field, providing a comprehensive understanding of linear conjugacy in algebraic structures.

Contribution

It offers a complete characterization of $Bbbk$-linear conjugacy for finite semigroups and groups over arbitrary fields, extending previous knowledge in algebra.

Findings

Provides criteria for $Bbbk$-linear conjugacy in finite semigroups.

Extends characterization to finite groups over any field.

Enhances understanding of conjugacy in linear representations.

Abstract

We say that two elements of a group or semigroup are $\Bbbk$-linear conjugates if their images under any linear representation over $\Bbbk$ are conjugate matrices. In this paper we characterize $\Bbbk$-linear conjugacy for finite semigroups (and, in particular, for finite groups) over an arbitrary field $\Bbbk$.