Assembling Crystals of Type A

Regular -crystals are certain edge-colored directed graphs, which are related to representations of the quantized universal enveloping algebra . For such a crystal with colors , we consider its maximal connected subcrystals with colors and with colors and characterize the interlacing structure for all pairs of these subcrystals. This enables us to give a recursive description of the combinatorial structure of via subcrystals and develop an efficient procedure of assembling . 1. Introduction Crystals are certain “exotic” edge-colored graphs. This graph-theoretic abstraction, introduced by Kashiwara [1, 2], has proved its usefulness in the theory of representations of Lie algebras and their quantum analogues. In general, a finite crystal is a finite directed graph such that the edges are partitioned into subsets, or color classes, labeled , each connected monochromatic subgraph of is a simple directed path, and there are certain interrelations between the lengths of such paths, which depend on the Cartan matrix related to a given Lie algebra . Of most interest are crystals of representations, or regular crystals. They are associated to elements of a certain basis of the highest weight integrable modules (representations) over the quantized universal enveloping algebra . This paper continues our combinatorial study of crystals begun in [3, 4] and considers -colored regular crystals of type A, where the number of colors is arbitrary. Recall that type A concerns ; in this case the Cartan matrix is viewed as if , if , and . We will refer to a regular -colored crystal of type A as an -crystal and omit the term when the number of colors is not specified. Since we are going to deal with finite regular crystals only, the adjectives “finite” and “regular” will usually be omitted. Also we assume that any crystal in question is (weakly) connected; that is, it is not the disjoint union of two nonempty graphs (which does not lead to loss of generality). It is known that any A-crystal possesses the following properties. (i) is acyclic (i.e., has no directed cycles) and has exactly one zero-indegree vertex, called the source, and exactly one zero-outdegree vertex, called the sink of . (ii) For any , each (inclusion-wise) maximal connected subgraph of whose edges have colors from is a crystal related to the corresponding submatrix of the Cartan matrix for . Throughout, speaking of a subcrystal of , we will mean a subgraph of this sort. Two-colored subcrystals are of especial importance, due to the result in [5] that for a crystal (of any type) with exactly one