2010/05/25 by Julianna Tymoczko, Tymoczko, Julianna · 6 citations
Mathematics · #05C10 #05E10 #Advanced Combinatorial Mathematics #Advanced Topics in Algebra #Algebraic structures and combinatorial models #Combinatorics (math.CO) #FOS: Mathematics
paper · pdf · doi:10.48550/arxiv.1005.4724
openalex publication_date 2010/05/25 · openalex created_date 2022/08/27 · openalex updated_date 2026/07/28
Combinatorial spiders are a model for the invariant space of the tensor\nproduct of representations. The basic objects, webs, are certain directed\nplanar graphs with boundary; algebraic operations on representations correspond\nto graph-theoretic operations on webs. Kuperberg developed spiders for rank 2\nLie algebras and sl2. Building on a result of Kuperberg's, Khovanov-Kuperberg\nfound a recursive algorithm giving a bijection between standard Young tableaux\nof shape (n,n,n) and irreducible webs for sl3 whose boundary vertices are all\nsources. In this paper, we give a simple and explicit map from standard Young\ntableaux of shape (n,n,n) to irreducible webs for sl3 whose boundary vertices\nare all sources, and show that it is the same as Khovanov-Kuperberg's map. Our\nconstruction generalizes to some webs with both sources and sinks on the\nboundary. Moreover, it allows us to extend the correspondence between webs and\ntableaux in two ways. First, we provide a short, geometric proof of\nPetersen-Pylyavskyy-Rhoades's recent result that rotation of webs corresponds\nto jeu-de-taquin promotion on (n,n,n) tableaux. Second, we define another\nnatural operation on tableaux called a shuffle, and show that it corresponds to\nthe join of two webs. Our main tool is an intermediary object between tableaux\nand webs that we call an m-diagram. The construction of m-diagrams, like many\nof our results, applies to shapes of tableaux other than (n,n,n).\n