Flag Hilbert schemes, colored projectors and Khovanov-Rozansky homology
Flag Hilbert schemes, colored projectors and Khovanov-Rozansky homology
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DOI：
10.1016/j.aim.2020.107542
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发表时间：
2016-08
影响因子：
1.7
通讯作者：
E. Gorsky;Andrei Neguct;J. Rasmussen
中科院分区：
文献类型：
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作者：
E. Gorsky;Andrei Neguct;J. Rasmussen
我们构造 A 型 Hecke 代数的最大交换子代数的分类。具体来说，我们提出了一个从标志希尔伯特方案上相干滑轮的（对称）幺半群范畴到Soergel 双模的（非对称）幺半群范畴的幺半群函子。该函子的伴随性允许将任何辫子的 Hochschild 同源性与标志希尔伯特方案上的束的欧拉特征相匹配。 Abel、Elias 和 Hogancamp 研究的分类 Jones-Wenzl 投影仪在 Soergel 双模范畴中是幂等的，它们对应于旗标 Hilbert 方案上环面不动点的重正化 Koszul 复形。因此，我们推测分类投影的自同态代数对应于标志希尔伯特方案的仿射图上函数的 dg 代数。我们在这些 dg 代数上定义了一系列微分 d N ，并推测它们的同源性与 gl N 投影仪的同源性相匹配，概括了第一作者和第三作者与 Oblomkov 和 Shende 的早期猜想。
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 N on these dg algebras and conjecture that their homology matches that of the gl N projectors, generalizing earlier conjectures of the first and third authors with Oblomkov and Shende.