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

In this paper we define an explicit basis for the $\mathfrak{sl}_n$-web algebra $H_n(\vec{k})$, the $\mathfrak{sl}_n$ generalization of Khovanov's arc algebra $H_{2}(m)$, using categorified $q$-skew Howe duality. Our construction is a $\mathfrak{sl}_n$-web version of Hu and 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 KL-R algebra and $H_n(\vec{k})$ and it gives an explicit graded cellular basis of the $2$-hom space between any two $\mathfrak{sl}_n$-webs $u$ and $v$. We use this to give a (in principle) computable version of colored Khovanov-Rozansky's $\mathfrak{sl}_n$-link homology. The complex we define is purely combinatorial and can be realized in the ("thick" cyclotomic) KL-R setting and needs only $F$'s. Moreover, we discuss some applications of our construction on the uncategorified level related to dual canonical bases of the $\mathfrak{sl}_n$-web space $W_n(\vec{k})$ and the MOY-calculus. Latter gives rise to a method to compute colored Reshetikhin-Turaev $\mathfrak{sl}_n$-link polynomials.