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Tropical subrepresentations of the boolean regular representation in low dimension

Steffen Marcus, Cameron Phillips

Abstract

We study two dimensional and three dimensional tropical subrepresentations of the regular representation $\mathbb{B}[G]$ of a finite group over the tropical booleans, utilizing the theory of group representations over a fixed idempotent semifield as developed by Giansiracusa--Manaker. In dimension two we completely classify all two dimensional tropical subrepresentations of $\mathbb{B}[G]$, provide an explicit characterization for the set of bases of the corresponding matroids, and show an equivalence with the subgroups of $G$. In dimension three we show such an equivalence no longer holds. Towards a classification in dimension three we give a collection of tropical subrepresentations corresponding to subgroups of index 2, and we show that in the special case of finite cyclic groups, one can find three dimensional tropical subrepresentations that do not correspond to subgroups in a similar way.

Tropical subrepresentations of the boolean regular representation in low dimension

Abstract

We study two dimensional and three dimensional tropical subrepresentations of the regular representation of a finite group over the tropical booleans, utilizing the theory of group representations over a fixed idempotent semifield as developed by Giansiracusa--Manaker. In dimension two we completely classify all two dimensional tropical subrepresentations of , provide an explicit characterization for the set of bases of the corresponding matroids, and show an equivalence with the subgroups of . In dimension three we show such an equivalence no longer holds. Towards a classification in dimension three we give a collection of tropical subrepresentations corresponding to subgroups of index 2, and we show that in the special case of finite cyclic groups, one can find three dimensional tropical subrepresentations that do not correspond to subgroups in a similar way.

Paper Structure

This paper contains 17 sections, 19 theorems, 66 equations.

Table of Contents

  1. Introduction
  2. Background
  3. The tropical booleans
  4. Exterior powers
  5. Tropical Linear Spaces and Matroids
  6. Linear representations and $G$-fixed Plücker vectors
  7. The regular representation
  8. 2-dimensional Subrepresentations of $\mathbb{B}[G]$
  9. Orbits and their properties
  10. Example: $\mathbb{Z}_n$
  11. Classification in dimension 2
  12. Examples
  13. 3-dimensional Subrepresentations
  14. Orbits and their properties
  15. Subrepresentations in dimension 3
  16. ...and 2 more sections

Key Result

Theorem 1

Let $G$ be a finite group. Two-dimensional tropical subrepresentations of $\mathbb{B}[G]$ correspond bijectively to proper subgroups $H\subset G$. The set of bases of the corresponding matroids are explicitly presented as a union of $G$-orbits for the induced $G$-action on subsets of $G$ size 2.

Theorems & Definitions (39)

  • Theorem : Theorem \ref{['thm:main']}
  • Theorem : Theorem \ref{['thm:dim3subgroups']}
  • Theorem : Theorem \ref{['thm:3d']}
  • Theorem 2.1: GM2020 Theorem A(1)
  • Proposition 3.1
  • proof
  • Proposition 3.2
  • proof
  • Proposition 3.3
  • proof
  • ...and 29 more