Categorified Young symmetrizers and stable homology of torus links

Abstract

We show that the triply graded Khovanov–Rozansky homology of the torus link $T_{n,k}$ stabilizes as $k\to \infty$. We explicitly compute the stable homology, as a ring, which proves a conjecture of Gorsky, Oblomkov, Rasmussen and 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.