The Robinson-Schensted Correspondence and A₂-web Bases

Authors

Matthew Housley
Heather M. Russell, University of Richmond
Julianna Tymoczko, Smith CollegeFollow

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 S_{3n-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.

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.

Recommended Citation

Housley, Matthew; Russell, Heather M.; and Tymoczko, Julianna, "The Robinson-Schensted Correspondence and A₂-web Bases" (2015). Mathematics Sciences: Faculty Publications, Smith College, Northampton, MA.
https://scholarworks.smith.edu/mth_facpubs/8