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The crossing model for regular $A_n$-crystals | | V.I.Danilov
; A.V.Karzanov
; G.A.Koshevoy
; | | Date: |
13 Dec 2006 | | Subject: | Representation Theory | | Abstract: | For a positive integer $n$, regular $A_n$-crystals are edge-colored directed graphs, with $n$ colors, related to integrable highest weight modules over the quantum algebra $U_q(sl_{n+1})$. Based on Stembridge’s local axioms for regular simply-laced crystals and a structural characterization of regular $A_2$-crystals in cite{DKK-06}, we introduce a new combinatorial construction, the so-called {em crossing model}, and prove that this model generates precisely the set of regular $A_n$-crystals. Using it, we obtain a series of results which significantly clarify the structure and demonstrate important ingredients of such crystals $K$. In particular, we reveal in $K$ a canonical subgraph called the skeleton and a canonical $n$-dimensional lattice $Pi$ of vertices and explain an interrelation of these objects. Also we show that there are exactly $ Pi $ maximal (connected) $A_{n-1}$-subcrystal $K’$ with colors $1,...,n-1$ (where neighboring colors do not commute) and that each $K’$ intersects $Pi$ at exactly one element, and similarly for the maximal subcrystals with colors $2,...,n$. | | Source: | arXiv, math/0612360 | | Services: | Forum | Review | PDF | Favorites | |
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