Assembling crystals of type A

Abstract

Regular $A_n$-crystals are certain edge-colored directed graphs which are related to representations of the quantized universal enveloping algebra $U_q(\mathfrak{sl}_{n+1})$. For such a crystal $K$ with colors $1,2,...,n$, we consider its maximal connected subcrystals with colors $1,...,n-1$ and with colors $2,...,n$ and characterize the interlacing structure for all pairs of these subcrystals. This is used to give a recursive description of the combinatorial structure of $K$ and develop an efficient procedure of assembling $K$.