Notes on Macdonald Polynomials and the Geometry of Hilbert Schemes

Curved Rickard complexes and link homologies

Journal für die reine und angewandte Mathematik (Crelles Journal) ◽

10.1515/crelle-2019-0044 ◽

2020 ◽

Vol 2020 (769) ◽

pp. 87-119

Author(s):

Sabin Cautis ◽

Aaron D. Lauda ◽

Joshua Sussan

Keyword(s):

Spectral Sequence ◽

Quantum Groups ◽

Braid Group ◽

Derived Categories ◽

Duke Math ◽

Khovanov Homology ◽

Hilbert Schemes ◽

Coherent Sheaves ◽

Knot Homology ◽

Original Motivation

AbstractRickard complexes in the context of categorified quantum groups can be used to construct braid group actions. We define and study certain natural deformations of these complexes which we call curved Rickard complexes. One application is to obtain deformations of link homologies which generalize those of Batson–Seed [3] [J. Batson and C. Seed, A link-splitting spectral sequence in Khovanov homology, Duke Math. J. 164 2015, 5, 801–841] and Gorsky–Hogancamp [E. Gorsky and M. Hogancamp, Hilbert schemes and y-ification of Khovanov–Rozansky homology, preprint 2017] to arbitrary representations/partitions. Another is to relate the deformed homology defined algebro-geometrically in [S. Cautis and J. Kamnitzer, Knot homology via derived categories of coherent sheaves IV, colored links, Quantum Topol. 8 2017, 2, 381–411] to categorified quantum groups (this was the original motivation for this paper).

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Evaluation of Nonsymmetric Macdonald Superpolynomials at Special Points

Symmetry ◽

10.3390/sym13050779 ◽

2021 ◽

Vol 13 (5) ◽

pp. 779

Author(s):

Charles F. Dunkl

Keyword(s):

Preceding Paper ◽

Hecke Algebra ◽

Type A ◽

Macdonald Polynomials ◽

Special Points ◽

Two Parameters

In a preceding paper the theory of nonsymmetric Macdonald polynomials taking values in modules of the Hecke algebra of type A (Dunkl and Luque SLC 2012) was applied to such modules consisting of polynomials in anti-commuting variables, to define nonsymmetric Macdonald superpolynomials. These polynomials depend on two parameters q,t and are defined by means of a Yang–Baxter graph. The present paper determines the values of a subclass of the polynomials at the special points 1,t,t2,… or 1,t−1,t−2,…. The arguments use induction on the degree and computations with products of generators of the Hecke algebra. The resulting formulas involve q,t-hook products. Evaluations are also found for Macdonald superpolynomials having restricted symmetry and antisymmetry properties.

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The close relation between border and Pommaret marked bases

Collectanea mathematica ◽

10.1007/s13348-021-00313-w ◽

2021 ◽

Author(s):

Cristina Bertone ◽

Francesca Cioffi

Keyword(s):

Finite Order ◽

Polynomial Ring ◽

Group Actions ◽

Close Relation ◽

Open Covering ◽

Order Ideal ◽

Lower Dimension ◽

Hilbert Schemes ◽

Affine Spaces ◽

The Ideal

AbstractGiven a finite order ideal $${\mathcal {O}}$$ O in the polynomial ring $$K[x_1,\ldots , x_n]$$ K [ x 1 , … , x n ] over a field K, let $$\partial {\mathcal {O}}$$ ∂ O be the border of $${\mathcal {O}}$$ O and $${\mathcal {P}}_{\mathcal {O}}$$ P O the Pommaret basis of the ideal generated by the terms outside $${\mathcal {O}}$$ O . In the framework of reduction structures introduced by Ceria, Mora, Roggero in 2019, we investigate relations among $$\partial {\mathcal {O}}$$ ∂ O -marked sets (resp. bases) and $${\mathcal {P}}_{\mathcal {O}}$$ P O -marked sets (resp. bases). We prove that a $$\partial {\mathcal {O}}$$ ∂ O -marked set B is a marked basis if and only if the $${\mathcal {P}}_{\mathcal {O}}$$ P O -marked set P contained in B is a marked basis and generates the same ideal as B. Using a functorial description of these marked bases, as a byproduct we obtain that the affine schemes respectively parameterizing $$\partial {\mathcal {O}}$$ ∂ O -marked bases and $${\mathcal {P}}_{\mathcal {O}}$$ P O -marked bases are isomorphic. We are able to describe this isomorphism as a projection that can be explicitly constructed without the use of Gröbner elimination techniques. In particular, we obtain a straightforward embedding of border schemes in affine spaces of lower dimension. Furthermore, we observe that Pommaret marked schemes give an open covering of Hilbert schemes parameterizing 0-dimensional schemes without any group actions. Several examples are given throughout the paper.

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Derived Hilbert schemes

Journal of the American Mathematical Society ◽

10.1090/s0894-0347-02-00399-5 ◽

2002 ◽

Vol 15 (4) ◽

pp. 787-815 ◽

Cited By ~ 15

Author(s):

Ionuţ Ciocan-Fontanine ◽

Mikhail M. Kapranov

Keyword(s):

Hilbert Schemes

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Transverse Hilbert schemes and completely integrable systems

Complex Manifolds ◽

10.1515/coma-2017-0015 ◽

2017 ◽

Vol 4 (1) ◽

pp. 263-272 ◽

Cited By ~ 1

Author(s):

Niccolò Lora Lamia Donin

Keyword(s):

Integrable Systems ◽

Hilbert Scheme ◽

Symplectic Form ◽

Initial Surface ◽

Hilbert Schemes ◽

Completely Integrable Systems ◽

Completely Integrable ◽

Symplectic Surface ◽

Complex Symplectic ◽

Minimum Polynomial

Abstract In this paper we consider a special class of completely integrable systems that arise as transverse Hilbert schemes of d points of a complex symplectic surface S projecting onto ℂ via a surjective map p which is a submersion outside a discrete subset of S. We explicitly endow the transverse Hilbert scheme Sp[d] with a symplectic form and an endomorphism A of its tangent space with 2-dimensional eigenspaces and such that its characteristic polynomial is the square of its minimum polynomial and show it has the maximal number of commuting Hamiltonians.We then provide the inverse construction, starting from a 2ddimensional holomorphic integrable system W which has an endomorphism A: TW → TW satisfying the above properties and recover our initial surface S with W ≌ Sp[d].

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NON-SYMPLECTIC INVOLUTIONS ON MANIFOLDS OF -TYPE

Nagoya Mathematical Journal ◽

10.1017/nmj.2019.43 ◽

2020 ◽

pp. 1-25

Author(s):

CHIARA CAMERE ◽

ALBERTO CATTANEO ◽

ANDREA CATTANEO

Keyword(s):

Moduli Spaces ◽

Quadratic Forms ◽

Symplectic Manifolds ◽

Hilbert Schemes ◽

Small Rank ◽

Twisted Sheaves ◽

Formula Type

We study irreducible holomorphic symplectic manifolds deformation equivalent to Hilbert schemes of points on a $K3$ surface and admitting a non-symplectic involution. We classify the possible discriminant quadratic forms of the invariant and coinvariant lattice for the action of the involution on cohomology and explicitly describe the lattices in the cases where the invariant lattice has small rank. We also give a modular description of all $d$ -dimensional families of manifolds of $K3^{[n]}$ -type with a non-symplectic involution for $d\geqslant 19$ and $n\leqslant 5$ and provide examples arising as moduli spaces of twisted sheaves on a $K3$ surface.

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