Linear dimension of group actions

TL;DR

This paper investigates the minimal dimension of linear representations that intertwine with permutation representations of groups, with applications in cryptography, by analyzing algebraic structures and specific classes of finite groups.

Contribution

It introduces a new algebraic approach to the linear dimension of group actions, providing structural results and explicit computations for various finite groups.

Findings

Computed linear dimensions for several finite primitive permutation groups

Determined linear dimensions for wreath products and imprimitive actions

Provided linear dimensions for almost simple finite 2-transitive groups

Abstract

Two fundamental ways to represent a group are as permutations and as matrices. In this paper, we study linear representations of groups that intertwine with a permutation representation. Recently, D'Alconzo and Di Scala investigated how small the matrices in such a linear representation can be. The minimal dimension of such a representation is the \emph{linear dimension of the group action} and this has applications in cryptography and cryptosystems. We develop the idea of linear dimension from an algebraic point of view by using the theory of permutation modules. We give structural results about representations of minimal dimension and investigate the implications of faithfulness, transitivity and primitivity on the linear dimension. Furthermore, we compute the linear dimension of several classes of finite primitive permutation groups. We also study wreath products, allowing us to determine the linear dimension of imprimitive group actions. Finally, we give the linear dimension of almost simple finite $2$-transitive groups, some of which may be used for further applications in cryptography. Our results also open up many new questions about linear representations of group actions.