${\mathfrak{sl}}_3$-web bases, intermediate crystal bases and categorification

Abstract

We give an explicit graded cellular basis of the ${\mathfrak{sl}}_3$-web algebra $K_S$. In order to do this, we identify Kuperberg's basis for the ${\mathfrak{sl}}_3$-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 ${\mathfrak{sl}}_3$-web algebra. We restrict ourselves to the ${\mathfrak{sl}}_3$ case over $\mathbb{C}$ here, but our approach should, up to the combinatorics of ${\mathfrak{sl}}_N$-webs, work for all $N>1$ or over $\mathbb{Z}$.

66 pages, lots of figures, some more typos fixed, added referee's suggestions, comments welcome

Cited by in corpus (4)

• Cellular structures using ${\text{U}}_q$-tilting modules
• ${\mathfrak{gl}}_n$-webs, categorification and Khovanov-Rozansky homologies
• Relative cellular algebras
• Real Springer fibers and odd arc algebras