String of columns rewriting and plactic-like monoids

par Nohra HAGE

jeudi 14 mai 2020, 14:00 → 15:00 Europe/Paris

112 virtuel (bât. Braconnier)

Lascoux and Schützenberger introduced the structure of plactic monoids after the works of Schensted and Knuth on the combinatorial study of Young tableaux. Using his jeu de taquin, Schützenberger gave the fi?rst correct proof of the Littelwood?-Richardsonrule. This rule describes in a combinatorial way the multiplicity of a Schur polynomial in
a product of Schur polynomials. Recently, similar classes of monoids such as hypoplactic,  Chinese, Sylvester and patience sorting monoids are also introduced and have found several applications on algebraic combinatorics and representation theory.

 

In this talk, we de?fine plactic-like monoids using the notions of string of columns constructed by insertion algorithms. We also introduce the notions of string data structures and morphism of string data structures and we explain how such a morphism can transfer combinatorial properties from a string data structure to another one, using sliding
algorithms and string of columns rewriting. Finally, we relate the string data structures of skew, Young and quasi-ribbon tableaux. In particular, we describe Schützenberger's jeu de taquin as a string of columns rewriting, showing that it is a morphism between the string data structures of skew and Young tableaux, using rewriting properties of the corresponding string of columns rewriting.