Flag Hilbert schemes, colored projectors and Khovanov-Rozansky homology
Name
1608.07308.pdf
Description
Submitted version
Size
1.18 MB
Format
Adobe PDF
Checksum (MD5)
c362cc5e1ed999e4d90c83d49a974da5
Author(s)
Gorsky, Eugene
Neguţ, Andrei
Rasmussen, Jacob
Date Issued
2021
Journal
Advances in Mathematics
Publisher
Elsevier BV
Version
Original manuscript
Abstract
We construct a categorification of the maximal commutative subalgebra of the type A Hecke algebra. Specifically, we propose a monoidal functor from the (symmetric) monoidal category of coherent sheaves on the flag Hilbert scheme to the (non-symmetric) monoidal category of Soergel bimodules. The adjoint of this functor allows one to match the Hochschild homology of any braid with the Euler characteristic of a sheaf on the flag Hilbert scheme. The categorified Jones-Wenzl projectors studied by Abel, Elias and Hogancamp are idempotents in the category of Soergel bimodules, and they correspond to the renormalized Koszul complexes of the torus fixed points on the flag Hilbert scheme. As a consequence, we conjecture that the endomorphism algebras of the categorified projectors correspond to the dg algebras of functions on affine charts of the flag Hilbert schemes. We define a family of differentials d on these dg algebras and conjecture that their homology matches that of the gl projectors, generalizing earlier conjectures of the first and third authors with Oblomkov and Shende. N N
MIT Department
Massachusetts Institute of Technology. Department of Mathematics
Terms of Use
Creative Commons Attribution-NonCommercial-NoDerivs License
Persistent DSpace Link
DOI of Published Version
https://doi.org/10.1016/j.aim.2020.107542
