A new tropical plactic algebra is introduced in which the Knuth relations are inferred from the underlying semiring arithmetic, encapsulating the ubiquitous plactic monoid Pn. This algebra manifests a natural framework for accommodating representations of Pn, and equivalently of Young tableaux, and its moderate coarsening — the cloaktic monoid Kn and the co-cloaktic monoid (Figure presented.). The faithful linear representations of Kn and (Figure presented.) by tropical matrices, which constitute a tropical plactic algebra, are shown to provide linear representations of the plactic monoid. To this end the paper develops a special type of configuration tableaux, corresponding bijectively to semi-standard Young tableaux. These special tableaux allow a systematic encoding of combinatorial properties in numerical algebraic ways, including algorithmic benefits. The interplay between these algebraic-combinatorial structures establishes a profound machinery for exploring semigroup attributes, in particular satisfying of semigroup identities. This machinery is utilized here to prove that Kn and (Figure presented.) admit all the semigroup identities satisfied by n×n triangular tropical matrices, which holds also for P3.
Izhakian, Z. (2019). Tropical plactic algebra, the cloaktic monoid, and semigroup representations. Journal of Algebra, 524, 290–366. https://doi.org/10.1016/j.jalgebra.2018.12.014