Tropical representations and identities of plactic monoids
Marianne Johnson, Mark Kambites · Transactions of the American Mathematical Society · 2021
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 Okniński 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 n is satisfied by the monoid of n × n n \times n upper triangular tropical matrices. In particular this implies that the variety generated by the 3 × 3 3 \times 3 upper triangular tropical matrices coincides with that generated by the plactic monoid of rank 3 3 , answering another question of Izhakian.