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 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 language | English |
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Pages (from-to) | 4423-4447 |
Journal | Transactions of the American Mathematical Society |
Volume | 374 |
Issue number | 6 |
Early online date | 30 Mar 2021 |
DOIs | |
Publication status | Published - 30 Mar 2021 |