Vladimir Shchigolev

Vladimir Shchigolev
Finance University under the Government of the Russian Federation · Department of Mathematics

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38
Publications
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201
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Publications

Publications (38)
Article
We construct combinatorial bases of the $T$-equivariant cohomology $H^\bullet_T(\Sigma,k)$ of the Bott-Samelson variety $\Sigma$ (where $T$ is the maximal torus) under some mild restrictions on the field of coefficients $k$. This bases allow us to prove the surjectivity of the restrictions $H^\bullet_T(\Sigma,k)\to H^\bullet_T(\pi^{-1}(x),k)$ and $...
Article
We consider Bott-Samelson varieties BSc(s) for a semisimple compact Lie group C corresponding to sequences of (not necessarily simple) reflections s. Let n be the lenth of s, K be a maximal torus in C and W be the Weyl group of C. For any set R of not overlapping integer pairs (i,j) such that 1⩽i⩽j⩽n and a function v:R→W, we consider the subspace B...
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We consider projection and lifting of labelled galleries to and from roots subsystems. Our constructions allow us to construct some topological embeddings of Bott-Samelson varieties skew equivariant with respect to the compact torus and order-preserving on the sets of points fixed by it.
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Let $T$ be a maximal torus of a semisimple complex algebraic group, $\mathrm{BS}(s)$ be the Bott-Samelson variety for a sequence of simple reflections $s$ and $\mathrm{BS}(s)^T$ be the set of $T$-fixed points of $\mathrm{BS}(s)$. We prove the tensor product decompositions for the image of the restriction $H^\bullet_T(\mathrm{BS}(s),k)\to H_T^\bulle...
Preprint
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We give a topological explanation of the main results of V.Shchigolev, Categories of Bott-Samelson Varieties, Algebras and Representation Theory, 23 (2), 349-391, 2020. To this end, we consider some subspaces of Bott-Samelson varieties invariant under the action of the maximal compact torus $K$ and study their topological and homological properties...
Article
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We consider all Bott-Samelson varieties BS(s) for a fixed connected semisimple complex algebraic group with maximal torus T as the class of objects of some category. The class of morphisms of this category is an extension of the class of canonical (inserting the neutral element) morphisms BS(s) → BS(s), where s is a subsequence of s. Every morphism...
Article
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We consider the quantum analog of the Lie commutator (Formula presented.) for an invertible element (Formula presented.) of the ground field and prove lower and upper bounds for the nilpotence degree of an associative algebra satisfying an identity of the form (Formula presented.).
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Let π:𝒩̃→𝒩 be the Springer resolution of the nilpotent cone for a semisimple connected algebraic group G over ℂ and k be an arbitrary field. What happens to π∗k̲𝒩̃[dim𝒩̃] if the decomposition theorem fails for it? We show that in this case, some additional (with respect to the case char k=0) composition factors of this direct image in the (abelian)...
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This paper is a complement to the work of the second author on modular quotient singularities in odd characteristic (see arXiv:1210.8006). Here we prove that if $V$ is a three-dimensional vector space over a field of characteristic $2$ and $G<GL(V)$ is a finite subgroup generated by pseudoreflections and possessing a $2$-dimensional invariant subsp...
Article
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Let $\pi:\widetilde{\mathcal N}\to\mathcal N$ be the Springer resolution of the nilpotent cone for a semisimple connected algebraic group $G$ over $\mathbb C$ and $k$ be an arbitrary field. What happens to $\pi_*k[\dim\mathcal N]$ if the decomposition theorem fails for it? We show that in this case, some additional (with respect to the case ${\rm...
Article
Let $\pi:\widetilde{\mathcal N}\to\mathcal N$ be the Springer resolution of the nilpotent cone for $\mathfrak{sl}_n(\mathbb C)$ and $k$ be a field. What happens to $\pi_*k[\dim\mathcal N]$ if the decomposition theorem fails for it? We show that in this case, some additional (with respect to the case ${\rm char}\;k$=0) composition factors of this di...
Article
We give an exact algorithm to calculate (under some GKM-restriction) the matrix describing the embedding B(s)(x) subset of B(s)(x), where the first module is the costalk and the second one is the stalk at x of a Bott-Samelson module (sheaf) B(s). This allows us to calculate the first few terms of the decomposition of B(s) into a sum of indecomposab...
Article
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There are two approaches to projective representation theory of symmetric and alternating groups, which are powerful enough to work for modular representations. One is based on Sergeev duality, which connects projective representation theory of the symmetric group and representation theory of the algebraic supergroup $Q(n)$ via appropriate Schur (s...
Article
Let G be a universal Chevalley group over an algebraically closed field and U− be the subalgebra of generated by all divided powers Xα,m with α<0. We conjecture an algorithm to determine if , where F∈U−, ω is a dominant weight and is a highest weight vector of the Weyl module Δ(ω). This algorithm does not use bases of Δ(ω) and is similar to the alg...
Article
Let F be an algebraically closed field of characteristic p>0. Suppose that SLn−1(F) is naturally embedded into SLn(F) (either in the top left corner or in the bottom right corner). We prove that certain Weyl modules over SLn−1(F) can be embedded into the restriction L(ω)↓SLn−1(F), where L(ω) is a simple SLn(F)-module. This allows us to construct ne...
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We consider the generalization of Kleshchev's lowering operators obtained by raising all the Carter-Lusztig operators in their definition to a power less than the characteristic of the ground field. If we apply such an operator to a nonzero GL_{n-1}-high weight vector of an irreducible representation of GL_n, shall we get a nonzero GL_{n-1}-high we...
Article
The modules and are described for certain simple -modules (the completely splittable ones or close to them), where is a field of characteristic and is the symmetric group of degree . This result is based on an upper bound for the dimensions of the -spaces between certain simple modules.
Article
In this paper, we find an explicit combinatorial criterion for the existence of a nonzero GL_{n-1}(K)-high weight vector of weight (\lambda_1,...,\lambda_{i-1},\lambda_i-d,\lambda_{i+1},..., \lambda_{n-1}), where d<char K and K is an algebraically closed filed, in the irreducible rational GL_n(K)-module L_n(\lambda_1,...,\lambda_n) with highest wei...
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In this paper, we calculate the space Ext GL(n 1 )(L n (λ), L n (μ)), where GL(n) is the general linear group of degree n over an algebraically closed field of positive characteristic, L n (λ) and L n (μ) are rational irreducible GL(n)-modules with highest weights λ and μ, respectively, the restriction of L n (λ) to any Levi subgroup of GL(n)...
Article
For an algebraically closed base field of positive characteristic, an algorithm to construct some non-zero GL ( n - 1 )-high weight vectors of irreducible rational GL ( n )-modules is suggested. It is based on the criterion proved in this paper for the existence of a set A such that Si, n ( A ) fμ, λ is a non-zero GL ( n - 1 )-high weight vector, w...
Article
The paper is a review of recent results on T-spaces and their applications. It is proved that T-spaces over a field of zero characteristic from a large class are finitely based; this implies the positive solution of the local Specht problem. If the characteristic of the base field is positive, then similar assertions do not hold. The negative solut...
Article
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New exact modular branching rules are obtained for modules over the symmetric groups that are close to completely splittable modules. These results are based on some upper bounds for the Ext^1-spaces between simple modules.
Article
We obtain a formula for \operatorname{Ext}^1_{K\Sigma_r}(D^\lambda,D^\mu) where K is a field of characteristic grater than 2, \Sigma_r is the symmetric group of degree r, D^\lambda and D^\mu are simple K\Sigma_r-modules, D^\lambda is a completely splittable module, and \lambda does not strictly dominate \mu. Bibtex entry for this abstract Preferred...
Article
We obtain a formula for Ext KΣr1 (D λ, D μ), where K is a field of characteristic grater than 2, Σ r is the symmetric group of degree r, D λ and D μ arc simple KΣ r- modules, D λis a completely splittable module, and λ does not strictly dominate μ.
Article
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In this paper we study the possibility to define irreducible representations of the symmetric groups with the help of finitely many relations. The existence of finite bases is established for the classes of representations corresponding to two-part partitions and to partitions from the fundamental alcove.
Article
A finite basis problem in Specht modules is considered. Some criteria are proved for a submodule to be generated by a submodule of the Specht module with fewer columns. Also a positive solution is obtained for two-row diagrams.
Article
In this paper we prove that any T-space over a field of characteristic zero has a finite basis. This result generalizes Kemer's theorem on the existence of a finite basis for any system of associative identities over a field of characteristic zero.
Article
This work is devoted to the construction of T-spaces with an infinite basis over a field of finite characteristic and over some other rings. Examples of T-spaces are given that are generated by polynomials in two variables or by polynomials of bounded degree in each variable.
Article
In the present article we consider finite basis property of some systems of commutative polynomials. In addition, we require sometimes that these systems are invariant under the action of the group GL∞(K). In this article we call such systems S-ideals. In the case of zero characteristic S-ideals are studied in [5]. Let K be a field, X (r) = {x i (j...

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