Tropical representations and identities of plactic monoids

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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$ 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.
Original languageEnglish
Pages (from-to)4423-4447
JournalTransactions of the American Mathematical Society
Issue number6
Early online date30 Mar 2021
Publication statusPublished - 30 Mar 2021


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