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Linear dimension of group actions

Alice Devillers, Michael Giudici, Daniel R. Hawtin, Lukas Klawuhn, Luke Morgan

TL;DR

This paper develops a unified algebraic framework for the linear dimension of group actions, conceptualizing minimal-dimensional representations that intertwine with a permutation action as quotients of permutation modules. By leveraging permutation-module theory, it provides precise LinDim values for broad families, including primitive and imprimitive actions, wreath products, and almost simple 2-transitive groups, with sharp results and constructive witnesses. The work both consolidates known results (e.g., on $S_n$ and $A_n$ actions) and advances new classifications via module-theoretic tools (such as the fully deleted permutation module and Brauer-tree analysis), offering cryptographic implications and a spectrum of natural open problems. Overall, it illuminates how the internal module structure of permutation representations governs the feasibility of compact linear actions in diverse group-action settings.

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.

Linear dimension of group actions

TL;DR

This paper develops a unified algebraic framework for the linear dimension of group actions, conceptualizing minimal-dimensional representations that intertwine with a permutation action as quotients of permutation modules. By leveraging permutation-module theory, it provides precise LinDim values for broad families, including primitive and imprimitive actions, wreath products, and almost simple 2-transitive groups, with sharp results and constructive witnesses. The work both consolidates known results (e.g., on and actions) and advances new classifications via module-theoretic tools (such as the fully deleted permutation module and Brauer-tree analysis), offering cryptographic implications and a spectrum of natural open problems. Overall, it illuminates how the internal module structure of permutation representations governs the feasibility of compact linear actions in diverse group-action settings.

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 -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.

Paper Structure

This paper contains 10 sections, 39 theorems, 90 equations, 1 table.

Table of Contents

  1. Introduction
  2. Representation theory preliminaries
  3. Group action representation preliminaries
  4. Quotients and a witness
  5. Intransitive actions
  6. Imprimitive actions
  7. Wreath products in product action
  8. Actions on sets and partitions
  9. Finite 2-transitive groups
  10. Some open problems

Key Result

Theorem 1.1

Let $G$ act primitively on $\Omega$ such that the induced permutation group $G^{\Omega}$ is not cyclic. Then $\mathop{\mathrm{LinDim}}\nolimits_F(G,\Omega)$ is the smallest codimension of a submodule of the permutation module $F^\Omega$ that is not a hyperplane. Moreover, if a witness for $\mathop{\

Theorems & Definitions (84)

  • Theorem 1.1
  • Theorem 1.2
  • Theorem 1.3
  • Theorem 1.4
  • Theorem 1.5
  • Theorem 2.1
  • proof
  • Lemma 3.1
  • proof
  • Lemma 3.2
  • ...and 74 more