Special Kirillov-Reshetikhin crystals

Abstract

We give a uniform realization of Kirillov-Reshetikhin crystals $B^{r,s}$, in terms of PBW crystals, for all affine types, $s\ge 1$, and nodes $r$ in the orbit of the $0$ node in the Dynkin diagram. Our proof is almost uniform except we prove some technical lemmas by first proving the case when the $r$-th fundamental weight ${\Lambda }_r$ is minuscule by using properties of minuscule crystal and then extend this to the case when ${\Lambda }_r$ is cominuscule using virtual crystals.

41 pages, 2 figures