Geodesic
$\frak{sl}_3$-web bases, intermediate crystal bases and categorification
Journal of Algebraic Combinatorics, Tome 40 (2014) no. 4, pp. 1001-1076.

Voir la notice de l'article provenant de la source Electronic Library of Mathematics

We give an explicit graded cellular basis of the $\frak{sl}_3$-web algebra $K_S$. In order to do this, we identify Kuperberg's basis for the $\frak{sl}_3$-web space $W_S$ with a version of Leclerc-Toffin's intermediate crystal basis and we identify Brundan, Kleshchev and Wang's degree of tableaux with the weight of flows on webs and the $q$-degree of foams. We use these observations to give a "foamy" version of Hu and Mathas graded cellular basis of the cyclotomic Hecke algebra which turns out to be a graded cellular basis of the $\frak{sl}_3$-web algebra. We restrict ourselves to the $\frak{sl}_3$ case over $\mathbb C$ here, but our approach should, up to the combinatorics of $\frak{sl}_N$-webs, work for all $N>1$ or over $\mathbb Z$.
Classification : 17B37, 20C08, 20C30
Keywords: categorification, q-skew Howe duality, webs and web-algebras, crystal bases, intermediate crystals 
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     title = {$\frak{sl}_3$-web bases, intermediate crystal bases and categorification},
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Tubbenhauer, Daniel. $\frak{sl}_3$-web bases, intermediate crystal bases and categorification. Journal of Algebraic Combinatorics, Tome 40 (2014) no. 4, pp. 1001-1076. http://geodesic.mathdoc.fr/item/JAC_2014__40_4_a4/