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Ilya Dumanski

Publications and source records attributed to Ilya Dumanski.

7 recordsLinked to original sources
arXiv2026

Noncommutative resolutions of affine Schubert varieties in type A and canonical bases

Given a resolution $\widetilde{\mathrm{Gr}}^{\underline{\lambda}} \rightarrow \overline{\mathrm{Gr}}^\lambda$ of an affine Schubert variety for $GL_n$, we define its noncommutative version -- a sheaf of algebras on $\overline{\mathrm{Gr}}^\lambda$, derived equivalent to $\widetilde{\mathrm{Gr}}^{\underline{\lambda}}$ as well as its Steinberg versions in both zero and positive characteristics. This, in particular, allows us to define the perversely-exotic t-structure on the derived category of equivariant coherent sheaves on $\widetilde{\mathrm{Gr}}^{\underline{\lambda}}$, analogously to Bezrukavnikov--Mirkovi\'c in the case of Springer resolution. We study the basis of classes of irreducible objects in the equivariant K-theory, and explicitly identify it with the (parabolic) Kazhdan--Lusztig canonical basis in a certain cell quotient. This allows us to relate it to the canonical basis for the quantum affine group. In the course of the proof, we establish some properties of coherent-constructible equivalences.

math.RT↗
arXiv2026

K-theoretic Hikita conjecture for quiver gauge theories

We study variants of Hikita conjecture for Nakajima quiver varieties and corresponding Coulomb branches. First, we derive the equivariant version of the conjecture from the non-equivariant one for a set of gauge theories. Second, we suggest a variant of the conjecture, with K-theoretic Coulomb branches involved. We show that this version follows from the usual (homological) one for a set of theories. We apply this result to prove the conjecture in finite ADE types. In the course of the proof, we show that appropriate completions of K-theoretic and homological (quantized) Coulomb branches are isomorphic.

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arXiv2025

Perverse coherent sheaves on symplectic singularities

We propose the notion of perverse coherent sheaves for symplectic singularities and study its properties. In particular, it gives a basis of simple objects in the Grothendieck group of Poisson sheaves. We show that perverse coherent bases for the nilpotent cone and for the affine Grassmannian arise as particular cases of our construction.

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arXiv2025

On reduced arc spaces of toric varieties

An arc space of an affine cone over a projective toric variety is known to be non-reduced in general. It was demonstrated recently that the reduced scheme structure is worth studying due to various connections with representation theory and combinatorics. In this paper we develop a general machinery for the description of the reduced arc spaces of affine cones over toric varieties. We apply our techniques to a number of classical cases and explore some connections with representation theory of current algebras.

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arXiv2024

A geometric approach to Feigin-Loktev fusion product and cluster relations in coherent Satake category

We propose a geometric realization of the Feigin-Loktev fusion product of graded cyclic modules over the current algebra. This allows us to compute it in several new cases. We also relate the Feigin-Loktev fusion product to the convolution of perverse coherent sheaves on the affine Grassmannian of the adjoint group. This relation allows us to establish the existence of exact triples, conjecturally corresponding to cluster relations in the Grothendieck ring of coherent Satake category.

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arXiv2022

Reduced arc schemes for Veronese embeddings and global Demazure modules

We consider arc spaces for the compositions of Pluecker and Veronese embeddings of the flag varieties for simple Lie groups of types ADE. The arc spaces are not reduced and we consider the homogeneous coordinate rings of the corresponding reduced schemes. We show that each graded component of a homogeneous coordinate ring is a cocyclic module over the current algebra and is acted upon by the algebra of symmetric polynomials. We show that the action of the polynomial algebra is free and that the fiber at the special point of a graded component is isomorphic to an affine Demazure module whose level is the degree of the Veronese embedding. In type A$_1$ (which corresponds to the Veronese curve) we give the precise list of generators of the reduced arc space. In general type, we introduce the notion of global higher level Demazure modules, which generalizes the standard notion of the global Weyl modules, and identify the graded components of the homogeneous coordinate rings with these modules.

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