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'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], 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 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'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 S3n-equivariant maps.
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The definitive version is available at: http://link.springer.com/article/10.1007/s10801-015-0582-5
Russell, Heather M., Matthew Housley, and Julianna Tymoczko. "The Robinson-Schensted Correspondence and A2-Web Bases."Journal of Algebraic Combinatorics 42, no. 1 (2015): 293-329. doi:10.1007/s10801-015-0582-5.
Russell, Heather M.; Housley, Matthew; and Tymoczko, Julianna, "The Robinson-Schensted Correspondence and A2-Web Bases" (2015). Math and Computer Science Faculty Publications. 126.