The Schützenberger groups and maximal subgroups of tropical matrices

TL;DR

This work provides a complete classification of the Schützenberger groups and maximal subgroups of tropical matrix structures. By leveraging semigroupoid language, automorphism descriptions of column/row spaces, and eigenvalue analyses, the authors show that Schützenberger groups of $M(\mathbb{T})$ (and hence $M(\mathbb{F}\mathbb{T})$) have a structured form built from real scalars and finite paired-2-closed permutation groups, assembled via wreath products; maximal subgroups of $M_n(\mathbb{T})$ follow a closely related but subtler pattern with rank constraints. They also provide a corrected proof for the maximal-subgroups classification, extend the results to the finitary tropical case, and demonstrate that some groups occur as Schützenberger groups without appearing as maximal subgroups for certain sizes $n$, highlighting nuanced differences between these two notions. The results give explicit descriptions in terms of internal semidirect products and wreath products, enriching the representation theory of tropical matrix semigroups and clarifying when a Schützenberger group can be realized as a maximal subgroup. Overall, the paper unifies the tropical-algebraic picture and resolves gaps in previous work, with concrete implications for $M_n(\mathbb{T})$ and $M_n(\mathbb{F}\mathbb{T})$.

Abstract

We classify the Schützenberger groups of the category of matrices over the tropical semiring, $M(\mathbb{T})$, in doing so, we obtain a classification for the Schützenberger groups of the semigroupoid of matrices over the finitary tropical semiring, $M(\mathbb{F}\mathbb{T})$. We then classify the maximal subgroups of the monoid of $n \times n$ matrices over the tropical semiring, $M_n(\mathbb{T})$, for all $n \in \mathbb{N}$; generalising a result in the literature and correcting an erroneous proof. We proceed to show that for some $n \in \mathbb{N}$ there exists a group which appears as a Schützenberger group of $M_n(\mathbb{T})$ but does not appear as a maximal subgroup.

Theorems & Definitions (65)

• Definition 2.1
• Definition 2.2
• Proposition 2.3
• proof
• Proposition 3.1
• proof
• Corollary 3.2
• Corollary 3.3
• Theorem 3.4
• proof