arXiv subjects

Galyna Dobrovolska

Publications and source records attributed to Galyna Dobrovolska.

12 recordsLinked to original sources

arXiv2023

On the structure of the affine asymptotic Hecke algebras

According to a conjecture of Lusztig, the asymptotic affine Hecke algebra should admit a description in terms of the Grothedieck group of sheaves on the square of a finite set equivariant under the action of the centralizer of a nilpotent element in the reductive group. A weaker form of this statement, allowing for possible central extensions of stabilizers of that action, has been proved by the first named author with Ostrik. In the present paper we describe an example showing that nontrivial central extensions do arise, thus the above weaker statement is optimal. We also show that Lusztig's homomorphism from the affine Hecke algebra to the asymptotic affine Hecke algebra induces an isomorphism on cocenters and discuss the relation of the above central extensions to the structure of the cocenter.

math.RT↗

arXiv2023

Combinatorial wall-crossing for the sign representation of prime size

We present an algorithm to calculate the result of combinatorial wall-crossing at every step starting with the column partition of prime size. This algorithm is confirmed by computer calculations. The output of the algorithm is consistent with Bezrukavnikov's conjecture in the case of the sign representation.

math.CO↗

arXiv2022

Crossingless sheaves and their classes in equivariant K-theory

We introduce crossingless sheaves in certain equivariant derived categories which are analogous to the Bezrukavnikov-Mirkovic exotic sheaves for two-block nilpotents. We calculate the classes of crossingless sheaves in equivariant K-theory of Cautis-Kamnitzer varieties.

math.AG↗

arXiv2022

Modular representations in type A with a two-row nilpotent central character

We study the category of representations of $\mathfrak{sl}_{m+2n}$ in positive characteristic, whose p-character is a nilpotent whose Jordan type is the two-row partition (m+n,n). In a previous paper with Anno, we used Bezrukavnikov-Mirkovic-Rumynin's theory of positive characteristic localization and exotic t-structures to give a geometric parametrization of the simples using annular crossingless matchings. Building on this, here we give combinatorial dimension formulae for the simple objects, and compute the Jordan-Holder multiplicities of the simples inside the baby Vermas (in special case where n=1, i.e. that a subregular nilpotent, these were known from work of Jantzen). We use Cautis-Kamnitzer's geometric categorification of the tangle calculus to study the images of the simple objects under the [BMR] equivalence. The dimension formulae may be viewed as a positive characteristic analogue of the combinatorial character formulae for simple objects in parabolic category O for $\mathfrak{sl}_{m+2n}$, due to Lascoux and Schutzenberger.

math.RT↗

arXiv2021

Some remarks on combinatorial wall-crossing

We establish a new simple explicit description of combinatorial wall-crossing for the rational Cherednik algebra applied to the trivial representation. In this way we recover a theorem of P. Dimakis and G. Yue. We also present two conjectures on combinatorial wall-crossing which were found using computer experiments.

math.CO↗

arXiv2018

Finite local systems in the Drinfeld-Laumon construction

Let E be a local system on a smooth projective curve of genus g with monodromy given by a representation of the symmetric group corresponding to a Young diagram with rows of lengths n_1,n_2,... where n_1 > n_2 + (2g-2), n_2 > n_3 + (2g-2), ..., n_{k-1} > n_k + (2g-2), n_k > n_{k+1} + n_{k+2} + ... + (2g-2). We show that the result of k steps of the Drinfeld-Laumon construction applied to E is the IC sheaf of the Harder-Narasimhan stratum with subquotients of rank 1 and degrees n_1, n_2, ..., n_k, n_{k+1}+n_{k+2}+... with coefficients in a local system with monodromy given by the Young diagram with rows n_{k+1}, n_{k+2}, ...

math.AG↗

arXiv2018

Towards a cluster structure on trigonometric zastava

We study a moduli problem on a nodal curve of arithmetic genus 1, whose solution is an open subscheme in the zastava space for projective line. This moduli space is equipped with a natural Poisson structure, and we compute it in a natural coordinate system. We compare this Poisson structure with the trigonometric Poisson structure on the transversal slices in an affine flag variety. We conjecture that certain generalized minors give rise to a cluster structure on the trigonometric zastava.

math.AG↗

arXiv2016

Moduli spaces, indecomposable objects and potentials over a finite field

Given a linear category over a finite field such that the moduli space of its objects is a smooth Artin stack (and some additional conditions) we give formulas for an exponential sum over the set of absolutely indecomposable objects and a stacky sum over the set of all objects of the category, respectively, in terms of the geometry of the cotangent bundle on the moduli stack. The first formula was inspired by the work of Hausel, Letellier, and Rodriguez-Villegas. It provides a new approach for counting absolutely indecomposable quiver representations, vector bundles with parabolic structure on a projective curve, and irreducible etale local systems (via a result of Deligne). Our second formula resembles formulas appearing in the theory of Donaldson-Thomas invariants.

math.AG↗

arXiv2015

Gaiotto-Witten superpotential and Whittaker D-modules on monopoles

Let $G$ be an almost simple simply connected group over complex numbers. For a positive element $α$ of the coroot lattice of $G$ let $Z^α$ denote the space of based maps from the projective line to the flag variety of $G$ of degree $α$. This space is known to be isomorphic to the space of framed euclidean $G$-monopoles with maximal symmetry breaking at infinity of charge $α$. In [Finkelberg-Kuznetsov-Markarian-Mirković] a system of (étale, rational) coordinates on $Z^α$ is introduced. In this note we compute various known structures on $Z^α$ in terms of the above coordinates. As a byproduct we give a natural interpretation of the Gaiotto-Witten superpotential and relate it to the theory of Whittaker D-modules introduced by D.Gaitsgory.

math.AG↗

arXiv2013

Fourier-Deligne transform and representations of the symmetric group

We calculate the Fourier-Deligne transform of the IC extension to ${\C}^{n+1}$ of the local system ${\mathcal L}_Λ$ on the cone over $\Conf_n(¶^1)$ associated to a representation $Λ$ of $S_n$, where the length $n-k$ of the first row of the Young diagram of $Λ$ is at least $\frac{|Λ|-1}{2}$. The answer is the IC extension to the dual vector space ${\C}^{n+1}$ of the local system ${\mathcal R}_λ$ on the cone over the $k$-th secant variety of the rational normal curve in $¶^n$, where ${\mathcal R}_λ$ corresponds to the representation $λ$ of $S_k$, the Young diagram of which is obtained from the Young diagram of $Λ$ by deleting its first row. We also prove an analogous statement for $S_n$-local systems on fibers of the Abel-Jacobi map. We use our result on the Fourier-Deligne transform to rederive a part of a result of Michel Brion on Kronecker coefficients.

math.AG↗

arXiv2012

On the lower central series of an associative algebra

This paper continues the study of the lower central series quotients of an associative algebra A, regarded as a Lie algebra, which was started in math/0610410 by Feigin and Shoikhet. Namely, it provides a basis for the second quotient in the case when A is the free algebra in n generators (note that the Hilbert series of this quotient was determined earlier in math/0610410). Further, it uses this basis to determine the structure of the second quotient in the case when A is the free algebra modulo the relations saying that the generators have given nilpotency orders. Finally, it determines the structure of the third and fourth quotient in the case of 2 generators, confirming an answer conjectured in math/0610410. Finally, in the appendix, the results of math/0610410 are generalized to the case when A is an arbitrary associative algebra (under certain conditions on $A$).

math.QA↗

arXiv2006

On products of sl_n characters and support containment

Let $λ$, $μ$, $ν$ and $ρ$ be dominant weights of $\mathfrak{sl_n}$ satisfying $λ+ μ= ν+ ρ$. Let $V_λ$ denote the highest weight module corresponding to $λ$. Lam, Postnikov, Pylyavskyy conjectured a sufficient condition for $V_λ \otimes V_μ$ to be contained in $V_ν \otimes V_ρ$ as $\mathfrak{sl_n}$-modules. In this note we prove a weaker version of the conjecture. Namely we prove that under the conjectured conditions every irreducible $\mathfrak{sl_n}$-module which appears in the decomposition of $V_λ \otimes V_μ$ does appear in the decomposition of $V_ν \otimes V_ρ$.

math.CO↗