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On the structure of regular $B_2$-type crystals | | V.I.Danilov
; A.V.Karzanov
; G.A.Koshevoy
; | | Date: |
21 Nov 2006 | | Subject: | Representation Theory; Combinatorics | | Abstract: | For simply-laced Kac-Moody algebras $frak g$, Stembridge (2003) proposed a `local’ axiomatization of crystal graphs of representations of $U_q(frak g)$. In this paper we propose axioms for edge-2-colored graphs which characterize the crystals of integrable representations of $U_q(sp(4))$, regular crystal graphs of $B_2$-type. An edge-colored directed graph which obeys our Axioms (K0)--(K5) is called an R-{em graph} (for brevity), and our main result is that the regular crystals of $B_2$-type are R-graphs and vice versa. We give a direct combinatorial construction for the crystals in question. On this way we introduce a new, so-called {em crossing model}, which does not exploit Young tableaux. This combinatorial model consists of a two-component graph of a rather simple form and of a certain set of integer-valued functions on its vertices. | | Source: | arXiv, math/0611641 | | Services: | Forum | Review | PDF | Favorites | |
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