Document Type
Article
Publication Date
8-2015
Publication Title
Journal of Algebraic Combinatorics
Abstract
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 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.
Keywords
𝔰𝔩(3) web basis, Kazhdan–Lusztig basis, Tau invariant Robinson–Schensted correspondence
Volume
42
Issue
1
First Page
293
Last Page
329
DOI
dx.doi.org/10.1007/s10801-015-0582-5
ISSN
1572-9192
Creative Commons License
This work is licensed under a Creative Commons Attribution 4.0 International License.
Rights
Licensed to Smith College and distributed CC-BY under the Smith College Faculty Open Access Policy. 12 month embargo at the request of the publisher.
Recommended Citation
Housley, Matthew; Russell, Heather M.; and Tymoczko, Julianna, "The Robinson-Schensted Correspondence and A2-web Bases" (2015). Mathematics and Statistics: Faculty Publications, Smith College, Northampton, MA.
https://scholarworks.smith.edu/mth_facpubs/8
Comments
Peer reviewed accepted manuscript. Language included at the request of the publisher: The final publication is available at Springer via http://dx.doi.org//10.1007/s10801-015-0582-5.
Full text also available at http://arxiv.org/pdf/1307.6487v2.pdf.