arXiv subjects

Rostislav Devyatov

Publications and source records attributed to Rostislav Devyatov.

11 recordsLinked to original sources

arXiv2025

Upper Bounds on the Torsion Index of Half-Spin Groups

The torsion index of split simple groups has been extensively studied, notably by Totaro, who calculated the torsion indexes of the spin groups and $E_{8}$ in [5] and [6], respectively. The aim of this paper is to provide upper bounds for the torsion index of half-spin groups, the only remaining case in the calculation of torsion indexes for split simple groups. We present general upper bounds for the torsion index of half-spin groups, showing that, except for certain exceptional cases, it is at most twice that of the corresponding spin groups. For these exceptional cases, the torsion index is bounded above by at most $2^3$ times that of the spin groups. Our results also reveal that in many cases, the torsion index of half-spin groups coincides with that of the spin groups.

math.AG↗

arXiv2023

Counter-examples to a conjecture of Karpenko for spin groups

Consider the canonical morphism from the Chow ring of a smooth variety $X$ to the associated graded ring of the topological filtration on the Grothendieck ring of $X$. In general, this morphism is not injective. However, Nikita Karpenko conjectured that these two rings are isomorphic for a generically twisted flag variety $X$ of a semisimple group $G$. The conjecture was first disproved by Nobuaki Yagita for $G=\mathop{\mathrm{Spin}}(2n+1)$ with $n=8, 9$. Later, another counter-example to the conjecture was given by Karpenko and the first author for $n=10$. In this note, we provide an infinite family of counter-examples to Karpenko's conjecture for any $2$-power integer $n$ greater than $4$. This generalizes Yagita's counter-example and its modification due to Karpenko for $n=8$.

math.AG↗

arXiv2020

An estimate of canonical dimension of groups based on Schubert calculus

We sketch the proof of a connection between the canonical (0-)dimension of semisimple split simply connected groups and cohomology of their full flag varieties. Using this connection, we get a new estimate of the canonical (0-)dimension of simply connected split exceptional groups of type $E$ understood as a group.

math.AG↗

arXiv2019

Oriented cohomology sheaves on double moment graphs

In the present paper we extend the theory of sheaves on moment graphs due to Braden-MacPherson and Fiebig to the context of an arbitrary oriented equivariant cohomology h (e.g. to algebraic cobordism). We introduce and investigate structure h-sheaves on double moment graphs to describe equivariant oriented cohomology of products of flag varieties. We show that in the case of a total flag variety X of Dynkin type A the space of global sections of the double structure h-sheaf also describes the endomorphism ring of the equivariant h-motive of X.

math.AG↗

arXiv2017

Multiplicity-free products of Schubert divisors

Let $G/B$ be a flag variety over an arbitrary field, where $G$ is a semisimple split algebraic group with a simply laced Dynkin diagram, and $B$ is a Borel subgroup. We say that the product of several classes of Schubert divisors in the Chow ring is \emph{multiplicity-free} if it is possible to multiply it by a Schubert class (not necessarily of a divisor) and get the class of a point. In the present paper we find all possible degrees (in the Chow ring) of multiplicity-free products of classes of Schubert divisors. Also, given a product of several classes of Schubert divisors, we can decompose it into a linear combination of classes of Schubert varieties with (as was known before) nonnegative coefficients. We study the coefficients in this linear combination and provide a criterion detecting if such a coefficient equals 1, is greater than 1, or equals zero (i.e. a Schubert variety is not actually present in the linear combination).

math.AG↗

arXiv2016

The K-theory of versal flags and cohomological invariants of degree 3

Let $G$ be a split semisimple linear algebraic group over a field and let $X$ be a generic twisted flag variety of $G$. Extending the Hilbert basis techniques to Laurent polynomials over integers we give an explicit presentation of the Grothendieck ring $K_0(X)$ in terms of generators and relations in the case $G=G^{sc}/μ_2$ is of Dynkin type ${\rm A}$ or ${\rm C}$ (here $G^{sc}$ is the simply-connected cover of $G$); we compute various groups of (indecomposable, semi-decomposable) cohomological invariants of degree 3, hence, generalizing and extending previous results in this direction.

math.AG↗

arXiv2015

Equivariant deformations of algebraic varieties with an action of an algebraic torus of complexity 1

Let $X$ be a 3-dimensional affine variety with a faithful action of a 2-dimensional torus $T$. Then the space of first order infinitesimal deformations $T^1(X)$ is graded by the characters of $T$, and the zeroth graded component $T^1(X)_0$ consists of all equivariant first order (infinitesimal) deformations. Suppose that using the construction of such varieties from [1], one can obtain $X$ from a proper polyhedral divisor $\mathscr D$ on $\mathbb P^1$ such that the tail cone of (any of) the used polyhedra is pointed and full-dimensional, and all vertices of all polyhedra are lattice points. Then we compute $\dim T^1(X)_0$ and find a formally versal equivariant deformation of $X$. We also establish a connection between our formula for $\dim T^1(X)_0$ and known formulas for the dimensions of the graded components of $T^1$ of toric varieties.

math.AG↗

arXiv2015

On Subword Complexity of Morphic Sequences

We study structure of pure morphic and morphic sequences and prove the following result: the subword complexity of arbitrary morphic sequence is either $Θ(n^{1+1/k})$ for some $k\in\mathbb N$, or is $O(n \log n)$.

math.CO↗

arXiv2013

Unipotent commutative group actions on flag varieties and nilpotent multiplications

Our goal is to classify all generically transitive actions of commutative unipotent groups on flag varieties up to conjugation. We establish relationship between this problem and classification of multiplications with certain properties on Lie algebra representations. Then we classify multiplications with the desired properties and solve the initial classification problem.

math.AG↗

arXiv2010

Generically transitive actions on multiple flag varieties

Let $G$ be a semisimple algebraic group whose decomposition into a product of simple components does not contain simple groups of type $A$, and $P\subseteq G$ be a parabolic subgroup. Extending the results of Popov [7], we enumerate all triples $(G, P, n)$ such that (a) there exists an open $G$-orbit on the multiple flag variety $G/P\times G/P\times\ldots\times G/P$ ($n$ factors), (b) the number of $G$-orbits on the multiple flag variety is finite.

math.AG↗

arXiv2009

Several examples of neigbourly polyhedra in co-dimension 4

In the article, a series of neigbourly polyhedra is constructed. They have $N=2d+4$ vertices and are embedded in $\mathbb R^{2d}$. Their (affine) Gale diagrams in $\mathbb R^2$ have $d+3$ black points that form a convex polygon. These Gale diagams can be enumerated using 3-trees (trees with some additional structure). Given $d$ and $m$, each of the constructed polyhedra in $\mathbb R^{2d}$ has a fixed number of faces of dimension $m$ that contain a vertex $A$. (This number depends on $d$ and $m$ does not depend on the polyhedron and the vertex $A$).

math.CO↗