Categorified Young symmetrizers and stable homology of torus links II

TL;DR

This paper constructs categorified Young symmetrizers using Soergel bimodules, proves a conjecture relating their Hochschild homology to the flag Hilbert scheme, and links this to Khovanov-Rozansky homologies of torus links.

Contribution

It introduces complexes of Soergel bimodules that categorify Young idempotents and proves a conjecture connecting their Hochschild homology with geometric and link homology structures.

Findings

Confirmed Gorsky-Rasmussen conjecture for $P_{1^n}$

Established a limit relation between Hochschild and Khovanov-Rozansky homologies

Derived new combinatorial results related to the homologies

Abstract

We construct complexes $P_{1^n}$ of Soergel bimodules which categorify the Young idempotents corresponding to one-column partitions. A beautiful recent conjecture of Gorsky-Rasmussen relates the Hochschild homology of categorified Young idempotents with the flag Hilbert scheme. We prove this conjecture for $P_{1^n}$ and its twisted variants. We also show that this homology is also a certain limit of Khovanov-Rozansky homologies of torus links. Along the way we obtain several combinatorial results which could be of independent interest.