A simple bijection between standard 3×n tableaux and irreducible webs for $mathfrak{sl}_{3}$

摘要

Combinatorial spiders are a model for the invariant space of the tensor product of representations. The basic objects, webs, are certain directed planar graphs with boundary; algebraic operations on representations correspond to graph-theoretic operations on webs. Kuperberg developed spiders for rank 2 Lie algebras and $mathfrak {sl}_{2}$ . Building on a result of Kuperberg, Khovanov–Kuperberg found a recursive algorithm giving a bijection between standard Young tableaux of shape 3×n and irreducible webs for $mathfrak{sl}_{3}$ whose boundary vertices are all sources.