The RobinsonSchensted correspondence and \(A_2\)-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 Kuperbergs 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 TemperleyLieb 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 RobinsonSchensted algorithm between permutations and Young tableaux and KhovanovKuperbergs bijection between Young tableaux and reduced webs. One main result uses Vogans 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 KazhdanLusztigleft 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 KhovanovKuperbergs bijection as an analogue of the RobinsonSchensted 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.
robinsonschensted correspondence \(\mathfrak {sl}(3)\) web basis kazhdanlusztig basis tau invariant