An explicit bijection between semistandard tableaux and non-elliptic sl 3 webs

Abstract

The sl 3 spider is a diagrammatic category used to study the representation theory of the quantum group U q (sl 3). The morphisms in this category are generated by a basis of non-elliptic webs. Khovanov–Kuperberg observe that non-elliptic webs are indexed by semistandard Young tableaux. They establish this bijection via a recursive growth algorithm. Recently, Tymoczko gave a simple version of this bijection in the case that the tableaux are standard and used it to study rotation and join of webs. This article builds on Tymoczko’s bijection to give a simple and explicit algorithm for constructing all non-elliptic sl 3 webs. As an application, we generalize results of Petersen–Pylyavskyy–Rhoades and Tymoczko proving that, for all non-elliptic sl 3 webs, rotation corresponds to jeu de taquin promotion and join corresponds to shuffling.