${\mathfrak{g}\mathfrak{l}}_n$-webs, categorification and Khovanov-Rozansky homologies

In this paper we define an explicit basis for the ${\mathfrak{g}\mathfrak{l}}_n$-web algebra $H_n(\vec{k})$ (the ${\mathfrak{g}\mathfrak{l}}_n$ generalization of Khovanov’s arc algebra) using categorified $q$-skew Howe duality. Our construction is a ${\mathfrak{g}\mathfrak{l}}_n$-web version of Hu–Mathas’ graded cellular basis and has two major applications: it gives rise to an explicit isomorphism between a certain idempotent truncation of a thick calculus cyclotomic KLR algebra and $H_n(\vec{k})$, and it gives an explicit graded cellular basis of the $2$-hom space between two ${\mathfrak{g}\mathfrak{l}}_n$-webs. We use this to give a (in principle) computable version of colored Khovanov-Rozansky ${\mathfrak{g}\mathfrak{l}}_n$-link homology, obtained from a complex defined purely combinatorially via the (thick cyclotomic) KLR algebra and needs only $F$.