The Robinson-Schensted Correspondence and $A_2$-web Bases

TL;DR

This paper explores the relationship between web bases and left cell bases in symmetric group representations, revealing structural connections via generalized tau-invariants and comparing classical bijections, while also establishing their fundamental differences.

Contribution

It introduces a new perspective on the Robinson-Schensted correspondence by relating it to Khovanov-Kuperberg's bijection and analyzes the structural similarities and differences between web and left cell bases.

Findings

Generalized tau-invariants are preserved under classical maps.

Khovanov-Kuperberg's bijection is an analogue of Robinson-Schensted.

Web and left cell bases are not equivalent under symmetric group actions.

Abstract

We study natural bases for two constructions of the irreducible representation of the symmetric group corresponding to $[n,n,n]$: the {\em reduced web} basis associated to Kuperberg's combinatorial description of the spider category; and the {\em left cell basis} for the left cell construction of Kazhdan and Lusztig. In the case of $[n,n]$, the spider category is the Temperley-Lieb category; reduced webs correspond to planar matchings, which are equivalent to left cell bases. This paper compares the images of these bases under classical maps: the {\em Robinson-Schensted algorithm} between permutations and Young tableaux and {\em Khovanov-Kuperberg's bijection} between Young tableaux and reduced webs. One main result uses Vogan's generalized $\tau$-invariant to uncover a close structural relationship between the web basis and the left cell basis. Intuitively, generalized $\tau$-invariants refine the data of the inversion set of a permutation. We define generalized $\tau$-invariants intrinsically for Kazhdan-Lusztig left cell basis elements and for webs. We then show that the generalized $\tau$-invariant is preserved by these classical maps. Thus, our result allows one to interpret Khovanov-Kuperberg's bijection as an analogue of the Robinson-Schensted correspondence. Despite all of this, our second main result proves that the reduced web and left cell bases are inequivalent; that is, these bijections are not $S_{3n}$-equivariant maps.