## The Robinson-Schensted correspondence and $A_2$-web bases

##### J. Algebr. Comb. 42(1), 293-329 (2015)
DOI: 10.1007/s10801-015-0582-5

### Summary

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 $\tau$-invariant to uncover a close structural relationship between the web basis and the left cell basis. Intuitively, generalized $\tau$-invariants refine the data of the inversion set of a permutation. We define generalized $\tau$-invariants intrinsically for Kazhdan-Lusztig left cell basis elements and for webs. We then show that the generalized $\tau$-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.

### Mathematics Subject Classification

05E10, 20B30, 20C30

### Keywords/Phrases

$\frak{sl}(3)$ web basis, Kazhdan-Lusztig basis, tau invariant, Robinson-Schensted correspondence