Tropical representations and semigroup identities of plactic-like monoids

This talk concerns several well-studied families of monoids whose elements are combinatorial gadgets of a particular type over a fixed finite alphabet, and whose multiplication can be defined by means of an insertion algorithm. The prototypical example is the family of finite rank plactic monoids: these are infinite monoids arising from a natural combinatorial multiplication (determined by Schensted’s insertion algorithm) on semistandard tableaux over a fixed finite alphabet. The finite rank plactic monoids, and many finite rank `plactic-like’ monoids, can be faithfully represented by matrices over the tropical semiring; we construct such representations and utilise these to study semigroup identities satisfied by the finite rank plactic (joint work with Kambites) and plactic-like (joint work with Cain, Kambites and Malheiro) monoids.