On the combinatorial structure of crystals of types A,B,C

Abstract

Regular $A_n$-, $B_n$- and $C_n$-crystals are edge-colored directed graphs, with ordered colors $1,2,...,n$, which are related to representations of quantized algebras $U_q({\mathfrak{sl}}_{n+1})$, $U_q({\mathfrak{sp}}_{2n})$ and $U_q({\mathfrak{so}}_{2n+1})$, respectively. We develop combinatorial methods to reveal refined structural properties of such objects. Firstly, we study subcrystals of a regular $A_n$-crystal $K$ and characterize pairwise intersections of maximal subcrystals with colors $1,...,n-1$ and colors $2,...,n$. This leads to a recursive description of the structure of $K$ and provides an efficient procedure of assembling $K$. Secondly, using merely combinatorial means, we demonstrate a relationship between regular $B_n$-crystals (resp. $C_n$-crystals) and regular symmetric $A_{2n-1}$-crystals (resp. $A_{2n}$-crystals).

54 pages. Compared with the previous version, some illustrations and references are added

Cited by in corpus (1)

• Assembling crystals of type A