Tropical representations and identities of plactic monoids

Mark Kambites, University of Manchester. Part of the Algebra, Logic, and Algorithms Seminar Series.

Plactic monoids are infinite, finitely generated monoids arising from a natural combinatorial multiplication on semistandard Young tableaux.  Introduced by Knuth and first systematically studied by Lascoux and Schutzenberger, they have many applications in algebraic combinatorics.  I shall discuss a recent discovery, made jointly with Marianne Johnson, of faithful representations of these monoids by matrices over the tropical semiring.  The existence of such representations (which answers a question of Izhakian) implies that plactic monoids satisfy semigroup identities (which was a conjecture of Kubat and Okninski).