Flag Hilbert schemes, colored projectors and Khovanov-Rozansky homology
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 adjo...
| Main Authors: | , , |
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| Format: | Article |
| Language: | English |
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Elsevier BV
2021
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| Online Access: | https://hdl.handle.net/1721.1/133980 |
| _version_ | 1826215362682683392 |
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| author | Gorsky, Eugene Neguţ, Andrei Rasmussen, Jacob |
| author2 | Massachusetts Institute of Technology. Department of Mathematics |
| author_facet | Massachusetts Institute of Technology. Department of Mathematics Gorsky, Eugene Neguţ, Andrei Rasmussen, Jacob |
| author_sort | Gorsky, Eugene |
| collection | MIT |
| description | 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 |
| first_indexed | 2024-09-23T16:25:52Z |
| format | Article |
| id | mit-1721.1/133980 |
| institution | Massachusetts Institute of Technology |
| language | English |
| last_indexed | 2024-09-23T16:25:52Z |
| publishDate | 2021 |
| publisher | Elsevier BV |
| record_format | dspace |
| spelling | mit-1721.1/1339802024-01-02T19:11:32Z Flag Hilbert schemes, colored projectors and Khovanov-Rozansky homology Gorsky, Eugene Neguţ, Andrei Rasmussen, Jacob Massachusetts Institute of Technology. Department of Mathematics 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 2021-10-27T19:57:29Z 2021-10-27T19:57:29Z 2021 2021-05-25T14:04:08Z Article http://purl.org/eprint/type/JournalArticle https://hdl.handle.net/1721.1/133980 en 10.1016/j.aim.2020.107542 Advances in Mathematics Creative Commons Attribution-NonCommercial-NoDerivs License http://creativecommons.org/licenses/by-nc-nd/4.0/ application/pdf Elsevier BV arXiv |
| spellingShingle | Gorsky, Eugene Neguţ, Andrei Rasmussen, Jacob Flag Hilbert schemes, colored projectors and Khovanov-Rozansky homology |
| title | Flag Hilbert schemes, colored projectors and Khovanov-Rozansky homology |
| title_full | Flag Hilbert schemes, colored projectors and Khovanov-Rozansky homology |
| title_fullStr | Flag Hilbert schemes, colored projectors and Khovanov-Rozansky homology |
| title_full_unstemmed | Flag Hilbert schemes, colored projectors and Khovanov-Rozansky homology |
| title_short | Flag Hilbert schemes, colored projectors and Khovanov-Rozansky homology |
| title_sort | flag hilbert schemes colored projectors and khovanov rozansky homology |
| url | https://hdl.handle.net/1721.1/133980 |
| work_keys_str_mv | AT gorskyeugene flaghilbertschemescoloredprojectorsandkhovanovrozanskyhomology AT negutandrei flaghilbertschemescoloredprojectorsandkhovanovrozanskyhomology AT rasmussenjacob flaghilbertschemescoloredprojectorsandkhovanovrozanskyhomology |