The Robinson–Schensted correspondence and $$A_2$$ A 2 -web bases

Abstract

We study natural bases for two constructions of the irreducible representation of the symmetric group corresponding to $$[n,n,n]$$[n,n,n]: the reduced web basis associated to Kuperberg's combinatorial description of the spider category; and the left cell basis for the left cell construction of Kazhdan and Lusztig. In the case of $$[n,n]$$[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'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 $$S_{3n}$$S3n-equivariant maps.