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arXiv · 1705.04596

A note on identities in plactic monoids and monoids of upper-triangular tropical matrices

Alan J. Cain·Georg Klein·Łukasz Kubat·António Malheiro·Jan Okniński
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Abstract

This paper uses the combinatorics of Young tableaux to prove the plactic monoid of infinite rank does not satisfy a non-trivial identity, by showing that the plactic monoid of rank $n$ cannot satisfy a non-trivial identity of length less than or equal to $n$. A new identity is then proven to hold for the monoid of $n \times n$ upper-triangular tropical matrices. Finally, a straightforward embedding is exhibited of the plactic monoid of rank $3$ into the direct product of two copies of the monoid of $3\times 3$ upper-triangular tropical matrices, giving a new proof that the plactic monoid of rank $3$ satisfies a non-trivial identity.

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Alan J. Cain, Georg Klein, Łukasz Kubat, António Malheiro, Jan Okniński. 2017-05-12. A note on identities in plactic monoids and monoids of upper-triangular tropical matrices. https://arxiv.org/abs/1705.04596

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