Categorified Young symmetrizers and stable homology of torus links

TL;DR

This paper proves the stabilization of triply graded Khovanov-Rozansky homology for torus links as one parameter grows, explicitly computing the stable homology and confirming a conjecture in the field.

Contribution

It constructs complexes of Soergel bimodules categorifying Young symmetrizers and shows they form stable limits of Rouquier complexes, leading to the computation of stable homology.

Findings

Stable homology of torus links is explicitly computed as a ring.

The stable homology confirms a conjecture by Gorsky-Oblomkov-Rasmussen-Shende.

Construction of complexes P_n categorifies Young symmetrizers and forms stable limits.

Abstract

We show that the triply graded Khovanov-Rozansky homology of the torus link $T_{n,k}$ stablizes as $k\to \infty$. We explicitly compute the stable homology (as a ring), which proves a conjecture of Gorsky-Oblomkov-Rasmussen-Shende. To accomplish this, we construct complexes $P_n$ of Soergel bimodules which categorify the Young symmetrizers corresponding to one-row partitions and show that $P_n$ is a stable limit of Rouquier complexes. A certain derived endomorphism ring of $P_n$ computes the aforementioned stable homology of torus links.