# Tropical plactic algebra, the cloaktic monoid, and semigroup representations

## TL;DR

This paper introduces a tropical plactic algebra and related monoids, providing new linear representations via tropical matrices, and develops combinatorial tableaux tools to analyze semigroup identities and algebraic properties.

## Contribution

It defines a tropical plactic algebra and monoids, linking combinatorial tableaux with algebraic representations, and proves these monoids satisfy identities of tropical triangular matrices.

## Key findings

- Faithful linear representations of monoids by tropical matrices.
- Development of configuration tableaux for encoding Young tableaux.
- Proof that monoids satisfy identities of tropical triangular matrices.

## Abstract

A new tropical plactic algebra is introduced in which the Knuth relations are inferred from the underlying semiring arithmetics, encapsulating the ubiquitous plactic monoid $\mathcal{P}_n$. This algebra manifests a natural framework for accommodating representations of $\mathcal{P}_n$, or equivalently of Young tableaux, and its moderate coarsening -- the cloaktic monoid $\mathcal{K}_n$ and the co-cloaktic $ ^{\operatorname{co}}\mathcal{K}_n$. The faithful linear representations of $\mathcal{K}_n$ and $\, ^{\operatorname{co}} \mathcal{K}_n$ by tropical matrices, which constitute a tropical plactic algebra, are shown to provide linear representations of the plactic monoid. To this end the paper develops a special type of configuration tableaux, corresponding bijectively to semi-standard Young tableaux. These special tableaux allow a systematic encoding of combinatorial properties in numerical algebraic ways, including algorithmic benefits. The interplay between these algebraic-combinatorial structures establishes a profound machinery for exploring semigroup attributes, in particular satisfying of semigroup identities. This machinery is utilized here to prove that $\mathcal{K}_n$ and $\, ^{\operatorname{co}} \mathcal{K}_n$ admit all the semigroup identities satisfied by $n \times n$ triangular tropical matrices, which holds also for $\mathcal{P}_3$.