Data

1. Title: $\mathfrak{gl}_n$-webs, categorification and Khovanov-Rozansky homologies
2. Author: Daniel Tubbenhauer
3. Status: To appear in J. Knot Theory Ramifications. Last update: Fri, 2 Oct 2020 05:17:24 EST
5. LaTex Beamer presentation: Slides1, Slides2, Slides3

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 $F$.

A few extra words

The main observation is that the $\mathfrak{gl}_n$-web space $W_n(\vec{k})$ is, under $q$-skew Howe duality, the $\dot{\textbf{U}}_q(\mathfrak{gl}_m)$-weight module of weight $\vec{k}$ in a certain $\dot{\textbf{U}}_q(\mathfrak{gl}_m)$-highest weight module.
It turns out that every $\mathfrak{gl}_n$-web can be obtained by a string of only $F$'s acting as elements of $\dot{\textbf{U}}_q(\mathfrak{gl}_m)$ on a highest weight vector. Thus, we can say that the $\mathfrak{gl}_n$-web space $W_n(\vec{k})$ is an instance of $\dot{\textbf{U}}_q(\mathfrak{sl}_m)$-highest weight theory.
A closed $\mathfrak{gl}_n$-web is therefore nothing else than a quantum number since one jumps from the $1$-dimensional highest weight space to the $1$-dimensional lowest weight space using the $F$'s of $\dot{\textbf{U}}_q(\mathfrak{sl}_m)$.
As a consequence: One can write each (colored) link as string of $F$'s together with operators $T$ for crossings. As for example displayed below for the Hopf link (with restriction to $n=2$). Here we act with the $F$'s from bottom to top and the $\dot{\textbf{U}}_q(\mathfrak{gl}_m)$-weight spaces can be read of from the number grid.
The operators $T$ measure the difference how one can hop around from the highest to the lowest weight and we obtain the Reshetikhin-Turaev $\mathfrak{gl}_n$-link polynomials therefore as an instance of $\dot{\textbf{U}}_q(\mathfrak{sl}_m)$-highest weight theory.
History repeats itself: The same is true for the $\mathfrak{gl}_n$-link homologies. We just replace the weight modules with weight categories, the $F$'s with functors and the measurement of the difference with certain natural transformations.
And since everything an the level of the $\mathfrak{gl}_n$-link polynomials takes place in a certain $\dot{\textbf{U}}_q(\mathfrak{sl}_m)$-module of highest weight we can obtain the Khovanov-Rozansky $\mathfrak{gl}_n$-link homologies by only using the cyclotomic KL-R algebra.
Thus, these homologies are instances of categorified $\dot{\textbf{U}}_q(\mathfrak{gl}_m)$-highest weight theory.