arXiv subjects

Vladimir Shchigolev

Publications and source records attributed to Vladimir Shchigolev.

At least 19 recordsLinked to original sources

arXiv2026

Homomorphisms between Bott-Samelson bimodules corresponding to sequences of reflections

We study the space of all bimodule homomorphisms $R_x\otimes_R R(\underline{t})\otimes_R R_y\to R_z\otimes_R R(\underline{t}')\otimes_R R_w$ as a one-sided module, where $R_x,R_y,R_z,R_w$ are standard twisted bimodules and $R(\underline{t})$ and $R(\underline{t}')$ are the Bott-Samelson bimodules corresponding to sequences of reflections $\underline{t}$ and $\underline{t}'$ respectively. We prove that this module is always reflexive under some reasonable restrictions on the representation of the underlying Coxeter group. However, unlike the case where $\underline{t}$ and $\underline{t}'$ contain only simple reflections, this module does not need any longer to be free. We provide a series of counterexamples already for the symmetric groups $S_n$, where $n\ge4$. The projective dimension of the modules dual to them is $n-3$ and thus serves to measure the deviation from the free modules. When placed within a geometric framework, these examples show how to find fibers of points fixed by the compact torus in the Bott-Samelson resolutions (as in the original definition by Raoul Bott and Hans Samelson) with non-vanishing odd cohomology.

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arXiv2025

Cycles in subexpression graphs

Let $\mathfrak{S}(\underline{s},w)$ be the graph whose vertices are all subexpressions with target $w$ of a fixed expression $\underline{s}$ in generators of a Coxeter group and edges are the pairs of subexpressions with Hamming distance 2. We prove that $\mathfrak{S}(\underline{s},w)$ is connected and its cycle space is spanned by cycles of lengths $d+2$, where $d$ ranges over all positive divisors of all finite orders of products of at most two entries of $\underline{s}$.

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arXiv2023

Twisted actions on cohomologies and bimodules

We introduce a twisted action of the equivariant cohomology of the singleton $H_R^\bullet({\rm pt},\Bbbk)$ on the equivarinat cohomology $H_L^\bullet(X,\Bbbk)$ of an $L$-space $X$. Considering this actions as a right action, $H_L^\bullet(X,\Bbbk)$ becomes a bimodule togeather with the canonical left action of $H_L^\bullet({\rm pt},\Bbbk)$. Using this bimodule structure, we prove an equivariant version of the Künneth isomorphism. We apply this result to the computation of the equivariant cohomologies of Bott-Samelson varieties and to a geometric construction of the bimodule morphisms between these cohomologies.

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arXiv2021

Galleries for root subsystems

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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arXiv2020

Tensor product decompositions for cohomologies of Bott-Samelson varieties

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 $H^\bullet_T(\mathrm{BS}(s),k)\to H_T^\bullet(X,k)$, where $X\subset\mathrm{BS}(s)^T$ is defined by some special not overlapping equations $γ_iγ_{i+1}\cdotsγ_j=w_{i,j}$ with right-hand sides belonging to the Weyl group.

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arXiv2020

Nested fibre bundles in Bott-Samelson varieties

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. Moreover, we describe multiplicative generators of the equivariant cohomologies of Bott-Samelson varieties.

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arXiv2017

Categories of Bott-Samelson varieties

We consider all Bott-Samelson varieties ${\rm BS}(s)$ 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 ${\rm BS}(s)\hookrightarrow{\rm BS}(s')$, where $s$ is a subsequence of $s'$. Every morphism of the new category induces a map between the $T$-fixed points but not necessarily between the whole varieties. We construct a contravariant functor from this new category to the category of graded $H^\bullet_T({\rm pt})$-modules coinciding on the objects with the usual functor $H_T^\bullet$ of taking $T$-equivariant cohomologies. We also discuss the problem how to define a functor to the category of $T$-spaces from a smaller subcategory. The exact answer is obtained for groups whose root systems have simply laced irreducible components by explicitly constructing morphisms between Bott-Samelson varieties (different from the canonical ones).

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arXiv2016

Three-dimensional isolated quotient singularities in even characteristic

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 subspace $W$ such that the restriction of $G$ to $W$ is isomorphic to the group $SL_{2}(\mathbb{F}_{2^n})$, then the quotient $V/G$ is non-singular. This, together with earlier known results on modular quotient singularities, implies first that a theorem of Kemper and Malle on irreducible groups generated by pseudoreflections generalizes to reducible groups in dimension three, and, second, that the classification of three-dimensional isolated singularities which are quotients of a vector space by a linear finite group reduces to Vincent's classification of non-modular isolated quotient singularities.

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arXiv2016

Bases of T-equivariant cohomology of Bott-Samelson varieties

We construct combinatorial bases of the $T$-equivariant ($T$ is the maximal torus) cohomology $H^\bullet_T(Σ,k)$ of the Bott-Samelson variety $Σ$ under some mild restrictions on the field of coefficients $k$. This bases allow us to prove the surjectivity of the restrictions $H^\bullet_T(Σ,k)\to H^\bullet_T(π^{-1}(x),k)$ and $H^\bullet_T(Σ,k)\to H^\bullet_T(Σ\setminusπ^{-1}(x),k)$, where $π:Σ\to G/B$ is the canonical resolution. In fact, we also construct bases of the targets of these restrictions by picking up certain subsets of certain bases of $H^\bullet_T(Σ,k)$ and restricting them to $π^{-1}(x)$ or $Σ\setminusπ^{-1}(x)$ respectively. As an application, we calculate the cohomology of the costalk-to-stalk embedding for the direct image $π_*{\underline k}_Σ$. This algorithm avoids division by 2, which allows us to reestablish 2-torsion for parity sheaves in Braden's example.

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arXiv2014

Composition factors for the Springer resolution

Let $π:\widetilde{\mathcal N}\to\mathcal N$ 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 $π_*k[\dim\mathcal N]$ 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 direct image in the (abelian) category of perverse sheaves may emerge. These factors emerge from the $Z_G(x)/Z_G(x)^0$-composition factors of the radicals of certain intersection forms and from that of the top comohologies of Springer fibres (in the non-semisimple case).

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arXiv2012

On decomposition of Bott-Samelson sheaves

We give an exact algorithm to calculate (under some GKM-restriction) the matrix describing the embedding $\B(\s)_x\subset\B(\s)^x$, 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 indecomposable modules (sheaves) and to calculate the characters of Braden--MacPherson sheaves in some previously unknown cases.

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arXiv2010

Modular branching rules for projective representations of symmetric groups and lowering operators for the supergroup Q(n)

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 (super)algebras and Schur functors. The second approach follows the work of Grojnowski for classical affine and cyclotomic Hecke algebras and connects projective representation theory of symmetric groups in characteristic $p$ to the crystal graph of the basic module of the twisted affine Kac-Moody algebra of type $A_{p-1}^{(2)}$. The goal of this work is to connect the two approaches mentioned above and to obtain new branching results for projective representations of symmetric groups. This is achieved by developing the theory of lowering operators for the supergroup $Q(n)$ which is parallel to (although much more intricate than) the similar theory for $GL(n)$ developed by the first author. The theory of lowering operators for $GL(n)$ is a non-trivial generalization of Carter's work in characteristic zero, and it has received a lot of attention. So this part of our work might be of independent interest. One of the applications of lowering operators is to tensor products of irreducible $Q(n)$-modules with natural and dual natural modules, which leads to important special translation functors. We describe the socles and primitive vectors in such tensor products.

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arXiv2009

Rectangular low level case of modular branching problem for GL_n(K)

In this paper, we find an explicit combinatorial criterion for the existence of a nonzero GL_{n-1}(K)-high weight vector of weight (λ_1,...,λ_{i-1},λ_i-d,λ_{i+1},..., λ_{n-1}), where d<char K and K is an algebraically closed filed, in the irreducible rational GL_n(K)-module L_n(λ_1,...,λ_n) with highest weight (λ_1,...,λ_n). For this purpose, new modular lowering operators are introduced.

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arXiv2009

Weyl submodules in restrictions of simple modules

Let F be an algebraically closed field of characteristic p>0. Suppose that SL_{n-1}(F) is naturally embedded into SL_n(F) (either in the top left corner or in the bottom right corner). We prove that certain Weyl modules over SL_{n-1}(F) can be embedded into the restriction L(ω)\downarrow_{SL_{n-1}(F)}, where L(ω) is a simple SL_n(F)-module. This allows us to construct new primitive vectors in L(ω)\downarrow_{\SL_{n-1}(F)} from any primitive vectors in the corresponding Weyl modules. Some examples are given to show that this result actually works.

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arXiv2009

A local criterion for Weyl modules for groups of type A

Let G be a universal Chevalley group over an algebraically closed field and U^- be the subalgebra of Dist(G) generated by all divided powers X_{α,m} with α<0. We conjecture an algorithm to determine if Fe^+_ω\ne0, where F\in\U^-, ωis a dominant weight and e^+_ωis a highest weight vector of the Weyl module Δ(ω). This algorithm does not use bases of Δ(ω) and is similar to the algorithm for irreducible modules that involves stepwise raising the vector under investigation. For an arbitrary G, this conjecture is proved in one direction and for G of type A in both.

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arXiv2007

Generalization of modular lowering operators for GL_n

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 weight vector again? The present paper gives the explicit answer to this question. In this way we obtain a new algorithm for generating some normal weights.

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arXiv2007

On some extensions of p-restricted completely splittable GL(n)-modules

In this paper, we calculate the space $Ext^1_{GL(n)}(L_n(λ),L_n(μ))$, 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) is semisimple, $λ$ is a $p$-restricted weight and $μ$ does not strictly dominate $λ$.

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arXiv2006

Iterating lowering operators

For an algebraically closed base field of characteristic p>0, a new algorithm to construct some non-zero GL(n-1)-high weight vectors of irreducible rational GL(n)-modules is suggested. It is based on successively applying Kleshchev's lowering operators to GL(n-1)-high weight vectors already obtained.

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