𝔤𝔩n-webs, categorification and Khovanov–Rozansky homologies

Abstract

In this paper, we define an explicit basis for the [Formula: see text]-web algebra [Formula: see text] (the [Formula: see text] generalization of Khovanov’s arc algebra) using categorified [Formula: see text]-skew Howe duality.Our construction is a [Formula: see text]-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 [Formula: see text], and it gives an explicit graded cellular basis of the [Formula: see text]-hom space between two [Formula: see text]-webs. We use this to give a (in principle) computable version of colored Khovanov–Rozansky [Formula: see text]-link homology, obtained from a complex defined purely combinatorially via the (thick cyclotomic) KLR algebra and needs only [Formula: see text].