2012/01/22 by Danilov, Vladimir, Karzanov, Alexander, Koshevoy, Gleb
#05C75 #05E99 #17B37 #Combinatorics (math.CO) #FOS: Mathematics
paper · doi:10.48550/arxiv.1201.4549
Regular An-, Bn- and Cn-crystals are edge-colored directed graphs, with ordered colors 1,2,...,n, which are related to representations of quantized algebras Uq(\mathfraksln+1), Uq(\mathfraksp2n) and Uq(\mathfrakso2n+1), respectively. We develop combinatorial methods to reveal refined structural properties of such objects. Firstly, we study subcrystals of a regular An-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 Bn-crystals (resp. Cn-crystals) and regular symmetric A2n-1-crystals (resp. A2n-crystals).