On the local structure of doubly laced crystals

Abstract

Let $\mathfrak{g}$ be a Lie algebra all of whose regular subalgebras of rank 2 are type $A_1\times A_1$, $A_2$, or $C_2$, and let $B$ be a crystal graph corresponding to a representation of $\mathfrak{g}$. We explicitly describe the local structure of $B$, confirming a conjecture of Stembridge.

16 pages, 7 figures First version used type $B_2$ crystals; current version uses type $C_2$ crystals. To appear in J. Comb. Theory, Ser. A