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Ilia Nekrasov

Publications and source records attributed to Ilia Nekrasov.

11 recordsLinked to original sources
arXiv2024

Upper bounds for measures on distal classes

In recent work, Harman and Snowden introduced a notion of measure on a Fraïssé class $\mathfrak{F}$, and showed how such measures lead to interesting tensor categories. Constructing and classifying measures is a difficult problem, and so far only a handful of cases have been worked out. In this paper, we obtain some of the first general results on measures. Our main theorem states that if $\mathfrak{F}$ is distal (in the sense of Simon), and there are some bounds on automorphism groups, then $\mathfrak{F}$ admits only finitely many measures; moreover, we give an effective upper bound on their number. For example, if $\mathfrak{F}$ is the class of ``$s$-dimensional permutations'' (finite sets equipped with $s$ total orders), we show that the number of measures is bounded above by approximately $\exp(\exp(s^2 \log{s}))$.

math.RT↗
arXiv2024

Explicit equations for exterior square of the general linear group

We present several explicit systems of equations defining exterior square of the general linear group as an affine group scheme. Algebraic ingredients of the equations, exterior numbers, are translated into the language of weight diagrams corresponding to Lie groups of type $A_{n-1}$ in representation with the highest weight $\varpi_{2}$.

math.GR↗
arXiv2024

Overgroups of exterior powers of an elementary group. Levels

We prove a first part of the standard description of groups $H$ lying between an exterior power of an elementary group $\bigwedge^m E_n(R)$ and a general linear group $GL_{n \choose m}(R)$ for a commutative ring $R$, $2\in R^*$ and $n\geqslant 3m$. The description uses the classical notion of a level: for every group $H$ we find a unique ideal $A$ of the ground ring $R$ which describes $H$.

math.GR↗
arXiv2024

Overgroups of exterior powers of an elementary group. Normalizers

We establish two characterizations of an algebraic group scheme $\bigwedge^m GL_n$ over $\mathbb{Z}$. Geometrically, the scheme $\bigwedge^m GL_n$ is a stabilizer of an explicitly given invariant form or, generally, an invariant ideal of forms. Algebraically, $\bigwedge^m GL_n$ is isomorphic (as a scheme over $\mathbb{Z}$) to a normalizer of the elementary subgroup functor $\bigwedge^m E_n$ and a normalizer of the subscheme $\bigwedge^m SL_n$. Our immediate goal is to apply both descriptions in the "sandwich classification" of overgroups of the elementary subgroup. Additionally, the results can be seen as a solution of the linear preserver problem for algebraic group schemes over $\mathbb{Z}$, providing a more functorial description that goes beyond geometry of the classical case over fields.

math.GR↗
arXiv2024

Arboreal tensor categories

We introduce some new symmetric tensor categories based on the combinatorics of trees: a discrete family $\mathcal{D}(n)$, for $n \ge 3$ an integer, and a continuous family $\mathcal{C}(t)$, for $t \ne 1$ a complex number. The construction is based on the general oligomorphic theory of Harman--Snowden, but relies on two non-trivial results we establish. The first determines the measures for the class of trees, and the second is a semi-simplicity theorem. These categories have some notable properties: for instance, $\mathcal{C}(t)$ is the first example of a 1-parameter family of pre-Tannakian categories of superexponential growth that cannot be obtained by interpolating categories of moderate growth.

math.RT↗
arXiv2020

Dual Infinite Wedge is $\mathrm{GL}_{\infty}$-equivariantly noetherian

We prove the (equivariant) noetherian property for a wide class of varieties generalizing the class of Plucker varieties (Theorem 1). It improves previous results of Draisma-Eggermont who treated the case of bounded Plucker varieties. Key ingredient of our proof is the constructive proof of the equivariant noetherianity for the hyper-Pfaffians (Theorem 36) which implies the equivariant noetherianity of the dual infinite wedge.

math.AG↗
arXiv2018

Compactifications of $M_{0,n}$ associated with Alexander self-dual complexes: Chow ring, $ψ$-classes and intersection numbers

An Alexander self-dual complex gives rise to a compactification of $M_{0,n}$, called ASD compactification, which is a smooth algebraic variety. ASD compactifications include (but are not exhausted by) the polygon spaces, or the moduli spaces of flexible polygons. We present an explicit description of the Chow rings of ASD compactifications. We study the analogues of Kontsevich tautological bundles, compute their Chern classes, compute top intersections of the Chern classes, and derive a recursion for the intersection numbers.

math.GT↗
arXiv2018

Overgroups of exterior powers of an elementary group. I. Levels and normalizers

In the present paper, we prove the first part in the standard description of groups $H$ lying between $m$-th exterior power of elementary group $E(n,R)$ and the general linear group $GL_{\binom{n}{m}}(R)$. We study structure of the exterior power of elementary group and its relative analog $E\left(\binom{n}{m},R,A\right)$. In the considering case $n \geq 3m$, the description is explained by the classical notion of level: for every such $H$ we find unique ideal $A$ of the ring $R$. Motivated by the problem, we prove the coincidence of the following groups: normalizer of the exterior power of elementary group, normalizer of the exterior power of special linear group, transporter of the exterior power of elementary group into the exterior power of special linear group, and an exterior power of general linear group. This result mainly follows from the found explicit equations for the exterior power of algebraic group scheme $GL_n(\_)$.

math.GR↗
arXiv2017

Intersection numbers of Chern classes of tautological line bundles on the moduli spaces of flexible polygons

Given a flexible $n$-gon with generic side lengths, the moduli space of its configurations in $\mathbb{R}^2$ as well as in $\mathbb{R}^3$ is a smooth manifold. It is equipped with $n$ \textit{tautological} line bundles whose definition is motivated by M. Kontsevich's tautological bundles over $\mathcal{M}_{0,n}$. We study their Euler classes, first Chern classes and intersection numbers, that is, top monomials in Chern (Euler) classes. The latter are interpreted geometrically as the signed numbers of some \textit{triangular configurations} of the flexible polygon.

math.GT↗
arXiv2016

Cyclopermutohedron: geometry and topology

The face poset of the permutohedron realizes the combinatorics of linearly ordered partitions of the set $[n]=\{1,...,n\}$. Similarly, the cyclopermutohedron is a virtual polytope that realizes the combinatorics of cyclically ordered partitions of the set $[n+1]$. The cyclopermutohedron was introduced by the third author by motivations coming from configuration spaces of polygonal linkages. In the paper we prove two facts: (1) the volume of the cyclopermutohedron equals zero, and (2) the homology groups $H_k$ for $k=0,...,n-2$ of the face poset of the cyclopermutohedron are non-zero free abelian groups. We also present a short formula for their ranks.

math.MG↗