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Abstract: Combinatorial spiders are a model for the invariant space of the tensorproduct of representations. The basic objects, webs, are certain directedplanar graphs with boundary; algebraic operations on representations correspondto graph-theoretic operations on webs. Kuperberg developed spiders for rank 2Lie algebras and sl 2. Building on a result of Kuperberg-s, Khovanov-Kuperbergfound a recursive algorithm giving a bijection between standard Young tableauxof shape n,n,n and irreducible webs for sl 3 whose boundary vertices are allsources. In this paper, we give a simple and explicit map from standard Youngtableaux of shape n,n,n to irreducible webs for sl 3 whose boundary verticesare all sources, and show that it is the same as Khovanov-Kuperberg-s map. Ourconstruction generalizes to some webs with both sources and sinks on theboundary. Moreover, it allows us to extend the correspondence between webs andtableaux in two ways. First, we provide a short, geometric proof ofPetersen-Pylyavskyy-Rhoades-s recent result that rotation of webs correspondsto jeu-de-taquin promotion on n,n,n tableaux. Second, we define anothernatural operation on tableaux called a shuffle, and show that it corresponds tothe join of two webs. Our main tool is an intermediary object between tableauxand webs that we call an m-diagram. The construction of m-diagrams, like manyof our results, applies to shapes of tableaux other than n,n,n.



Autor: Julianna Tymoczko

Fuente: https://arxiv.org/



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