# Tropical representations and identities of plactic monoids

## TL;DR

This paper provides a faithful tropical matrix representation of finite rank plactic monoids, proving they satisfy non-trivial identities and characterizing their algebraic variety through upper triangular tropical matrices.

## Contribution

It introduces a faithful tropical matrix representation for all finite rank plactic monoids and establishes their identities and algebraic varieties.

## Key findings

- Plactic monoids of finite rank can be faithfully represented as upper triangular tropical matrices.
- Every finite rank plactic monoid satisfies a non-trivial semigroup identity.
- The variety generated by 3x3 upper triangular tropical matrices matches that of the rank 3 plactic monoid.

## Abstract

We exhibit a faithful representation of the plactic monoid of every finite rank as a monoid of upper triangular matrices over the tropical semiring. This answers a question first posed by Izhakian and subsequently studied by several authors. A consequence is a proof of a conjecture of Kubat and Okni\'{n}ski that every plactic monoid of finite rank satisfies a non-trivial semigroup identity. In the converse direction, we show that every identity satisfied by the plactic monoid of rank $n$ is satisfied by the monoid of $n \times n$ upper triangular tropical matrices. In particular this implies that the variety generated by the $3 \times 3$ upper triangular tropical matrices coincides with that generated by the plactic monoid of rank $3$, answering another question of Izhakian.