The combinatorics of \(\mathsf{A}_2\)-webs

Georgia Benkart¹ ; Soojin Cho² ; Dongho Moon³

¹University of Wisconsin-Madison
²Ajou University
³Sejong University

The electronic journal of combinatorics, Tome 21 (2014) no. 2

Résumé

The nonelliptic $\mathsf{A_2}$-webs with $k$ "$+$''s on the top boundary and $3n-2k$ "$-$''s on the bottom boundary combinatorially model the space $\mathsf{Hom}_{\mathfrak{sl}_3}(\mathsf{V}^{\otimes (3n-2k)}, \mathsf{V}^{\otimes k})$ of $\mathfrak{sl}_3$-module maps on tensor powers of the natural 3-dimensional $\mathfrak{sl}_3$-module $\mathsf{V}$, and they have connections with the combinatorics ofSpringer varieties. Petersen, Pylyavskyy, and Rhodes showed that the set of such $\mathsf{A_2}$-webs and the set of semistandard tableaux of shape $(3^n)$ and type $\{1^2,\dots,k^2,k+1,\dots, 3n-k\}$ have the same cardinalities. In this work, we use the $\sf{m}$-diagrams introduced by Tymoczko and the Robinson-Schensted correspondence to construct an explicit bijection, different from the one given by Russell, between these two sets. In establishing our result, we show that the pair of standard tableaux constructed using the notion of path depth is the same as the pair constructed from applying the Robinson-Schensted correspondence to a $3\,2\,1$-avoiding permutation. We also obtain a bijection between such pairs of standard tableaux and Westbury's $\mathsf{A_2}$ flow diagrams.

Détail

DOI : 10.37236/3936

Classification : 05E10, 05A05, 05A19
Mots-clés : \(m\)-diagrams, Springer varieties, semistandard tableaux