Matthew Housley ; Heather M. Russell ; Julianna Tymoczko - The Robinson―Schensted Correspondence and $A_2$-webs

dmtcs:2349 - Discrete Mathematics & Theoretical Computer Science, January 1, 2013, DMTCS Proceedings vol. AS, 25th International Conference on Formal Power Series and Algebraic Combinatorics (FPSAC 2013) - https://doi.org/10.46298/dmtcs.2349
The Robinson―Schensted Correspondence and $A_2$-websArticle

Authors: Matthew Housley 1; Heather M. Russell 2; Julianna Tymoczko 3

The $A_2$-spider category encodes the representation theory of the $sl_3$ quantum group. Kuperberg (1996) introduced a combinatorial version of this category, wherein morphisms are represented by planar graphs called $\textit{webs}$ and the subset of $\textit{reduced webs}$ forms bases for morphism spaces. A great deal of recent interest has focused on the combinatorics of invariant webs for tensors powers of $V^+$, the standard representation of the quantum group. In particular, the invariant webs for the 3$n$th tensor power of $V^+$ correspond bijectively to $[n,n,n]$ standard Young tableaux. Kuperberg originally defined this map in terms of a graphical algorithm, and subsequent papers of Khovanov–Kuperberg (1999) and Tymoczko (2012) introduce algorithms for computing the inverse. The main result of this paper is a redefinition of Kuperberg's map through the representation theory of the symmetric group. In the classical limit, the space of invariant webs carries a symmetric group action. We use this structure in conjunction with Vogan's generalized tau-invariant and Kazhdan–Lusztig theory to show that Kuperberg's map is a direct analogue of the Robinson–Schensted correspondence.


Volume: DMTCS Proceedings vol. AS, 25th International Conference on Formal Power Series and Algebraic Combinatorics (FPSAC 2013)
Section: Proceedings
Published on: January 1, 2013
Imported on: November 21, 2016
Keywords: Robinson―Schensted,Web basis,Kazhdan―Lusztig theory,Young tableau,[INFO.INFO-DM] Computer Science [cs]/Discrete Mathematics [cs.DM]

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