Algebra and Logic Seminar - 11/11/2015

Wednesday, 11 November 2015, 2:30 p.m.

Lecturer: Alan Cain (CMA,FCT-UNL)

Title: "Crystals, Young tableau, and the (hypo)plactic monoid"

Local: Sala de Seminários, Edifício VII

Faculdade de Ciências e Tecnologia, Quinta da Torre, Caparica

Abstract: The plactic monoid (the monoid of Young tableaux) is closely connected with representations of the special linear Lie algebra and with the theory of symmetric functions. In particular, the representation-theoretic notions of Kashiwara operators and crystal bases can be applied to the plactic monoid in a purely combinatorial and monoid-theoretical way, with a very elegant interaction between the resulting crystal structure and the algebraic and combinatorial properties of the plactic monoid. Indeed, one can view the crystal structure as an alternative definition of the plactic monoid.

Krob & Thibon showed that the hypoplactic monoid (the monoid of quasi-ribbon tableaux, and a quotient of the plactic monoid) has a role for quasi-symmetric functions that is analoguous to the role of the plactic monoid for symmetric functions. However, there was no natural crystal structure known for the hypoplactic monoid.

This seminar will describe recent joint work with António Malheiro, in which we detach the notion of Kashiwara operators from the underlying representation theory, and introduce a notion of "quasi-Kashiwara operators" that give rise to a "quasi-crystal" structure for the hypoplactic monoid. This quasi-crystal structure leads to new results and improved proofs for known results.  It also illuminates on the relationship between the plactic monoid, the hypoplactic monoid, and the sylvester monoid (the monoid of binary search trees).

The exposition will be elementary. Representation theory and the theory of (quasi-)symmetric functions will only appear for motivation; no special knowledge of these areas will be assumed.