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

TL;DR

This paper constructs an explicit basis for the $rak{gl}_n$-web algebra using categorified duality, enabling computable colored Khovanov-Rozansky homology through combinatorial methods.

Contribution

It introduces a $rak{gl}_n$-web basis and establishes isomorphisms with cyclotomic KLR algebras, advancing the computational approach to link homology.

Findings

Explicit basis for $rak{gl}_n$-web algebra constructed.

Isomorphism with cyclotomic KLR algebra established.

Provides a combinatorial method for computing $rak{gl}_n$-link homology.

Abstract

In this paper we define an explicit basis for the $\mathfrak{gl}_n$-web algebra $H_n(\vec{k})$ (the $\mathfrak{gl}_n$ generalization of Khovanov's arc algebra) using categorified $q$-skew Howe duality. Our construction is a $\mathfrak{gl}_n$-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 $\mathfrak{gl}_n$-webs. We use this to give a (in principle) computable version of colored Khovanov-Rozansky $\mathfrak{gl}_n$-link homology, obtained from a complex defined purely combinatorially via the (thick cyclotomic) KLR algebra and needs only $F$.