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Pavel Shlykov

Publications and source records attributed to Pavel Shlykov.

4 recordsLinked to original sources
arXiv2026

K-theory of Gieseker variety and type A cyclotomic Hecke algebra

We give an algebraic description of the equivariant K-theory of Gieseker varieties. Our main result identifies the equivariant K-theory of the Gieseker space with the Jucys--Murphy center of the cyclotomic Hecke algebra, over the equivariant K-theory of a point. The construction is inspired by the proof of the Hikita--Nakajima conjecture for Gieseker spaces given by the first and third authors. We discuss consequences for the center of cyclotomic Hecke algebras and for specializations to q=1 and to roots of unity. In particular, we relate K-theory of affine type A quiver varieties with the centers of the corresponding blocks of specialized cyclotomic Hecke algebras. This last result strengthens the correspondences obtained by the second author in earlier work.

math.AG↗
arXiv2025

Nakajima quiver varieties in dimension four

This paper classifies all 4d Nakajima quiver varieties through a combinatorial approach. For each such variety, we describe the symplectic leaves and minimal degenerations between them. Using the resulting Hasse diagrams and secondary hyperplane arrangements, we fully classify the quiver varieties up to isomorphism, a step in the problem of classifying all 4d conical symplectic singularities and the (2, 2) case of quiver varieties. As an application, we answer in the negative a question posed by Bellamy, Craw, Rayan, Schedler, and Weiss regarding whether the $G_4$ quotient singularity (or its projective crepant resolutions) can be realised as a quiver variety.

math.AG↗
arXiv2023

Hikita-Nakajima conjecture for the Gieseker variety

Let $\mathfrak{M}_0$ be an affine Nakajima quiver variety, and $\mathcal{M}$ is the corresponding BFN Coulomb branch. Assume that $\mathfrak{M}_0$ can be resolved by the (smooth) Nakajima quiver variety $\mathfrak{M}$. The Hikita-Nakajima conjecture claims that there should be an isomorphism of (graded) algebras $H^*_{S}(\mathfrak{M},\mathbb{C}) \simeq \mathbb{C}[\mathcal{M}_{\mathfrak{s}}^{\mathbb{C}^\times}]$, here $S \curvearrowright \mathfrak{M}_0$ is a torus acting on $\mathfrak{M}_0$ preserving the Poisson structure, $\mathcal{M}_{\mathfrak{s}}$ is the (Poisson) deformation of $\mathcal{M}$ over $\mathfrak{s}=\operatorname{Lie} (S)$, $\mathbb{C}^\times$ is a generic one-dimensional torus acting on $\mathcal{M}$, and $\mathbb{C}[\mathcal{M}_{\mathfrak{s}}^{\mathbb{C}^\times}]$ is the algebra of schematic $\mathbb{C}^\times$-fixed points of $\mathcal{M}_{\mathfrak{s}}$. We prove the Hikita-Nakajima conjecture for $\mathfrak{M}=\mathfrak{M}(n,r)$ Gieseker variety ($ADHM$ space). We produce the isomorphism explicitly on generators. We also describe the Hikita-Nakajima isomorphism above using the realization of $\mathcal{M}_{\mathfrak{s}}$ as the spectrum of the center of rational Cherednik algebra corresponding to $S_n \ltimes (\mathbb{Z}/r\mathbb{Z})^n$ and identify all the algebras that appear in the isomorphism with the center of degenerate cyclotomic Hecke algebra (generalizing some results of Shan, Varagnolo, and Vasserot).

math.RT↗