The Robinson-Schensted Correspondence and A2-webs
Archived as published. Open access article.
Abstract
The A2-spider category encodes the representation theory of the sl3 quantum group. Kuperberg (1996) introduced a combinatorial version of this category, wherein morphisms are represented by planar graphs called webs and the subset of reduced webs forms bases for morphism spaces. A great deal of recent interest has focused on the combinatorics of invariant webs for tensors powers of V +, the standard representation of the quantum group. In particular, the invariant webs for the 3nth tensor power of V + correspond bijectively to [n, n, n] standard Young tableaux. Kuperberg originally defined this map in terms of a graphical algorithm, and subsequent papers of Khovanov-Kuperberg (1999) and Tymoczko (2012) introduce algorithms for computing the inverse. The main result of this paper is a redefinition of Kuperberg's map through the representation theory of the symmetric group. In the classical limit, the space of invariant webs carries a symmetric group action. We use this structure in conjunction with Vogan's generalized tau-invariant and Kazhdan-Lusztig theory to show that Kuperberg's map is a direct analogue of the Robinson-Schensted correspondence.