Schensted type correspondence for type $G_{2}$ and computation of the canonical basis of a finite dimensional $U_{q}(G_{2})$-module

Abstract

We use Kang-Misra's combinatorial description of the crystal graphs for $U_{q}(G_{2})$ to introduce the plactic monoid for type $G_{2}$. Then we describe the corresponding insertion algorithm which yields a Schensted type correspondence. Next we give a simple algorithm for computing the canonical basis of any finite dimensional $U_{q}(G_{2})$-module.