Tropical plactic algebra, the cloaktic monoid, and semigroup representations

A new tropical plactic algebra is introduced in which the Knuth relations are inferred from the underlying semiring arithmetics, encapsulating the ubiquitous plactic monoid Pn. This algebra manifests a natural framework for accommodating representations of Pn, or equivalently of Young tableaux, and its moderate coarsening -- the cloaktic monoid Kn and the co-cloaktic coKn. The faithful linear representations of Kn and coKn 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 coKn admit all the semigroup identities satisfied by n×n triangular tropical matrices, which holds also for P3.