arXiv:1212.5771v1 [math.CO] 23 Dec 2012 Assembling crystals of type A 1 Vladimir I. Danilov 2 , Alexander V. Karzanov 34 , and Gleb A. Koshevoy 5 Abstract. Regular A n -crystals are certain edge-colored directed graphs which are related to representations of the quantized universal enveloping algebra U q (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. Keywords : Crystals of representations, Simply laced Lie algebras AMS Subject Classification 17B37, 05C75, 05E99 1 Introduction Crystals are certain “exotic” edge-colored graphs. This graph-theoretic abstraction, introduced by Kashiwara [6, 7], has proved its usefulness in the theory of represen- tations of Lie algebras and their quantum analogues. In general, a finite crystal is a finite directed graph K such that: the edges are partitioned into n subsets, or color classes, labeled 1,...,n, each connected monochromatic subgraph of K is a simple directed path, and there is a certain interrelation between the lengths of such paths, which depends on the n × n Cartan matrix M =(m ij ) related to a given Lie al- gebra g. 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 U q (g). This paper continues our combinatorial study of crystals begun in [1, 2] and con- siders n-colored regular crystals of type A, where the number n of colors is arbitrary. Recall that type A concerns g = sl n+1 ; in this case the Cartan matrix M is viewed as m ij = −1 if |i − j | = 1, m ij = 0 if |i − j | > 1, and m ii = 2. We will refer to a regular 1 Supported by RFBR grant 10-01-9311-CNRSL a. 2 Central Institute of Economics and Mathematics of the RAS (47, Nakhimovskii Prospect, 117418 Moscow, Russia); email: danilov@cemi.rssi.ru. 3 Institute for System Analysis of the RAS (9, Prospect 60 Let Oktyabrya, 117312 Moscow, Russia); email: sasha@cs.isa.ru. 4 A part of this research was done while this author was visiting Equipe Combinatoire et Optimi- sation, Univ. Paris-6, and Institut f¨ ur Diskrete Mathematik, Univ. Bonn. Corresponding author. 5 Central Institute of Economics and Mathematics of the RAS (47, Nakhimovskii Prospect, 117418 Moscow, Russia) and Laboratoire J.-V.Poncelet (11, Bolshoy Vlasyevskiy Pereulok, 119002 Moscow, Russia); email: koshevoy@cemi.rssi.ru. 1