https://hal.inria.fr/hal-01229662Housley, MatthewMatthewHousleyBYU - Brigham Young UniversityRussell, Heather M.Heather M.RussellUSC - University of Southern CaliforniaTymoczko, JuliannaJuliannaTymoczkoSmith College [Northampton]The Robinson―Schensted Correspondence and $A_2$-websHAL CCSD2013Robinson―SchenstedWeb basisKazhdan―Lusztig theoryYoung tableau[INFO.INFO-DM] Computer Science [cs]/Discrete Mathematics [cs.DM]Monteil, AlainAlain Goupil and Gilles Schaeffer2015-11-17 10:19:302019-11-18 12:12:022015-11-17 10:25:14enConference papershttps://hal.inria.fr/hal-01229662/document10.46298/dmtcs.2349application/pdf1The $A_2$-spider category encodes the representation theory of the $sl_3$ quantum group. Kuperberg (1996) introduced a combinatorial version of this category, wherein morphisms are represented by planar graphs called $\textit{webs}$ and the subset of $\textit{reduced webs}$ 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 3$n$th 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.