A Plactic Algebra Action on Bosonic Particle Configurations

Abstract

AbstractWe study an action of the plactic algebra on bosonic particle configurations. These particle configurations together with the action of the plactic generators can be identified with crystals of the quantum analogue of the symmetric tensor representations of the special linear Lie algebra $\mathfrak {s} \mathfrak {l}_{N}$ s l N . It turns out that this action factors through a quotient algebra that we call partic algebra, whose induced action on bosonic particle configurations is faithful. We describe a basis of the partic algebra explicitly in terms of a normal form for monomials, and we compute the center of the partic algebra.