The Robinson-Schensted Correspondence and A2-Web Bases

We study natural bases for two constructions of the irreducible representation of the symmetric group corresponding to [n; n; n]: the reduced web basis associated to Kuperberg\u27s combinatorial description of the spider category; and the 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 image of these bases under classical maps: the Robinson-Schensted algorithm between permutations and Young tableaux and Khovanov-Kuperberg\u27s bijection between Young tableaux and reduced webs. One main result uses Vogan\u27s generalized T-invariant to uncover a close structural relationship between the web basis and the left cell basis. Intuitively, generalized T-invariants refine the data of the inversion set of a permutation. We define generalized T-invariants intrinsically for Kazhdan-Lusztig left cell basis elements and for webs. We then show that the generalized T-invariant is preserved by these classical maps. Thus, our result allows one to interpret Khovanov-Kuperberg\u27s 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 S3n-equivariant maps