2013/07/24 by Matthew Housley, Housley, Matthew, Heather M. Russell +3
Mathematics · #05E10 #05E15 #20C08 #20C30 #20G42 #57M27 #Advanced Algebra and Geometry #Advanced Combinatorial Mathematics #Algebraic structures and combinatorial models #Combinatorics (math.CO) #FOS: Mathematics #Geometric Topology (math.GT) #Representation Theory (math.RT)
paper · pdf · doi:10.48550/arxiv.1307.6487
openalex publication_date 2013/07/24 · openalex created_date 2025/10/10 · openalex updated_date 2026/07/28
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 τ-invariant to uncover a close structural relationship between the web basis and the left cell basis. Intuitively, generalized τ-invariants refine the data of the inversion set of a permutation. We define generalized τ-invariants intrinsically for Kazhdan-Lusztig left cell basis elements and for webs. We then show that the generalized τ-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.