Semigroup identities and varieties of plactic monoids

We study the semigroup identities satisfied by finite rank plactic monoids. We find a new set of semigroup identities of the plactic monoid of rank $n$ for $n \geq 4$, which are shorter than those previously known when $n \geq 6$. Using these semigroup identities we show that for all $n \in \mathbb{N}$, the plactic monoid of rank $n$ satisfies a semigroup identity not satisfied by the semigroup of $(n+1) \times (n+1)$ upper triangular tropical matrices. We then prove that the plactic monoid of rank $n$ generates a different semigroup variety for each rank $n$.