%0 Journal Article
%T Assembling Crystals of Type A
%A Vladimir I. Danilov
%A Alexander V. Karzanov
%A Gleb A. Koshevoy
%J Algebra
%D 2013
%I Hindawi Publishing Corporation
%R 10.1155/2013/483949
%X 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
%U http://www.hindawi.com/journals/algebra/2013/483949/