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Сложность действия редуктивных групп над алгебраически незамкнутым полем и сильная стабильность действий на флаговых многообразиях
Доклады Российской академии наук. Математика, информатика, процессы управления (ранее - Доклады Академии Наук. Математика). 2019. Т. 485. № 1.
Zhgoon V., Кноп Ф.
In press
We annonce the results generalizing the Vinberg's Complexity Theorem for the action of reductive group on an algebraic variety over algebraically non-closed field. Also we give new results on the strong $k$-stability for the actions on flag varieties.
Smirnov E., В кн. : Тезисы докладов шестой школы-конференции "Алгебры Ли, алгебраические группы и теория инвариантов". : МЦНМО, 2017. С. 74-75.
We give a new proof of a criterion for Kawamata log-terminality of a pair (G/P,D), where D is an effective divisor on a generalized flag variety G/P. ...
Added: October 7, 2019
Valentina Kiritchenko, Mathematical Research Letters 2016 Vol. 23 No. 4 P. 1069-1096
We describe an elementary convex geometric algorithm for realizing Schubert cycles in complete flag varieties by unions of faces of polytopes. For GL_n and Gelfand{Zetlin polytopes, combinatorics of this algorithm coincides with that of the mitosis on pipe dreams introduced by Knutson and Miller. For Sp_4 and a Newton{Okounkov polytope of the symplectic flag variety, ...
Added: February 25, 2016
Roman Avdeev, Petukhov A., Transformation Groups 2021 Vol. 26 No. 3 P. 719-774
Let G be a symplectic or special orthogonal group, let H be a connected reductive subgroup of G, and let X be a flag variety of G. We classify all triples (G, H, X) such that the natural action of H on X is spherical. For each of these triples, we determine the restrictions to ...
Added: September 2, 2020
Kiritchenko V., / Cornell University. Series math "arxiv.org". 2014.
We describe an elementary convex geometric algorithm for realizing Schubert cycles in complete flag varieties by unions of faces of polytopes. For GL_n and Gelfand--Zetlin polytopes, combinatorics of this algorithm coincides with that of the mitosis on pipe dreams introduced by Knutson and Miller. For Sp_4 and a Newton--Okounkov polytope of the symplectic flag variety, ...
Added: September 17, 2014
Leonid Monin, Smirnov E., Seminaire Lotharingien de Combinatoire 2023 Vol. 89B Article 76
In 1992, Pukhlikov and Khovanskii provided a description of the cohomology ring of toric variety as a quotient of the ring of differential operators on spaces of virtual polytopes. Later Kaveh generalized this construction to the case of cohomology rings for full flag varieties.
In this paper we extend Pukhlikov--Khovanskii type presentation to the case of K-theory ...
Added: October 26, 2023
Р.С. Авдеев, Петухов А. В., Математический сборник 2014 Т. 205 № 9 С. 3-48
For every finite-dimensional vector space V and every V-flag variety X we list all connected reductive subgroups in GL(V) acting spherically on X. ...
Added: October 22, 2014
Roman Avdeev, Petukhov A., Algebras and Representation Theory 2020 Vol. 23 No. 3 P. 541-581
Let G be a connected semisimple algebraic group and let H⊂G be a connected reductive subgroup. Given a flag variety X of G, a result of Vinberg and Kimelfeld asserts that H acts spherically on X if and only if for every irreducible representation R of G realized in the space of sections of a ...
Added: February 11, 2019
Smirnov E., , in : Recent Developments in Representation Theory. Vol. 673.: Providence : AMS, 2016. P. 179-226.
The aim of these notes is to give an introduction into Schubert calculus on Grassmannians and flag varieties. We discuss various aspects of Schubert calculus, such as applications to enumerative geometry, structure of the cohomology rings of Grassmannians and flag varieties, Schur and Schubert polynomials. We conclude with a survey of results of V. Kiritchenko, V. Timorin ...
Added: October 13, 2015
Smirnov E., Пенков И., Игнатьев М. В. et al., М. : ВИНИТИ РАН, 2018
Сборник трудов семинара по алгебре и геометрии Самарского государственного университета ...
Added: August 19, 2018
Smirnov E., / Cornell University. Series math "arxiv.org". 2016. No. 1609.07771.
We give an alternative proof of a recent result by Pasquier stating that for a generalized flag variety X=G/P and an effective Q-divisor D stable with respect to a Borel subgroup the pair (X,D) is Kawamata log terminal if and only if [D]=0. ...
Added: September 27, 2016
Zhgoon V., Труды научно-исследовательского института системных исследований Российской академии наук 2017 Т. 7 № 3 С. 57-60
В работе описана новая конструкция, дающая вложение специального вида много- образия полных флагов произвольной редуктивной группы. А именно, для произволь- ной редуктивной группы строятся вложения реализующие многообразие обобщенных полных флагов в качестве орбиты цепочки попарно вложенных подпространств спе- циального вида, граф вложения которых является диаграммой Дынкина, снабженной дополнительными данными. Эта конструкция дает новые реализации вложений ...
Added: October 11, 2017
Feigin E., Finkelberg M. V., Reineke M., / Cornell University. Series math "arxiv.org". 2014. No. 1410.0777.
We study the connection between the affine degenerate Grassmannians in type $A$, quiver Grassmannians for one vertex loop quivers and affine
Schubert varieties. We give an explicit description of the degenerate affine Grassmannian of type $GL_n$ and identify it
with semi-infinite orbit closure of type $A_{2n-1}$. We show that principal quiver Grassmannians for the one vertex loop ...
Added: October 6, 2014
Smirnov E., Bulletin of the Korean Mathematical Society 2017 Vol. 54 No. 5 P. 1773-1778
We give an alternative proof of a recent result by Pasquier stating that for a generalized flag variety X=G/P and an effective Q-divisor D stable with respect to a Borel subgroup the pair (X,D) is Kawamata log terminal if and only if [D]=0. ...
Added: February 14, 2017
Feigin E., Functional Analysis and Its Applications 2014 Vol. 48 No. 1 P. 59-71
The degenerate Lie group is a semidirect product of the Borel subgroup with the normal
abelian unipotent subgroup.
We introduce a class of the highest weight representations of the degenerate group of type A, generalizing
the PBW-graded representations of the classical group. Following the classical construction
of the flag varieties, we consider the closures of the orbits of the ...
Added: April 30, 2014
Smirnov E., Journal of Mathematical Sciences 2020 Vol. 248 No. 3 P. 338-373
This paper is a review of results on multiple flag varieties, i.e., varieties of the form G/P1×· · ·×G/Pr. We provide a classification of multiple flag varieties of complexity 0 and 1 and results on the combinatorics and geometry of B-orbits and their closures in double cominuscule flag varieties. We also discuss questions of finiteness for the ...
Added: July 6, 2020
Tyurin N. A., Математические заметки 2014 Т. 96 № 3 С. 476-479
Краткое сообщение ...
Added: January 21, 2015
И. В. Аржанцев, Ю. И. Зайцева, Успехи математических наук 2022 Т. 77 № 4(466) С. 3-90
Работа содержит обзор недавних результатов об открытых вложениях аффинного пространства C^n в полные алгебраические многообразия X, для которых действие векторной группы G_a^n на C^n параллельными переносами продолжается до действия G_a^n на X. В первой части работы мы подробно изучаем соответствие Хассетта–Чинкеля, описывающее эквивариантные вложения C^n в проективные пространства, и приводим его обобщение на случай вложений в проективные гиперповерхности. Последующие разделы посвящены изучению вложений в многообразия флагов и в их вырождения, в полные торические ...
Added: August 4, 2022
Ignatyev Mikhail V., Shevchenko A., Bochkarev M., Journal of Algebra 2016 Vol. 465 P. 259-286
We study tangent cones to Schubert subvarieties of the flag variety of a complex reductive group G. Let T be a maximal torus of G, B be a Borel subgroup of G containing T, Φ be the root system of G with respect to T, W be the Weyl group of Φ, and F = G/B be the flag variety. We prove that if every irreducible component of Φ is of type Bn or ...
Added: October 11, 2023
Merzon G., Smirnov E., / Cornell University. Series math "arxiv.org". 2014. No. 1410.6857.
We prove new determinantal identities for a family of flagged Schur polynomials. As a corollary of these identities we obtain determinantal expressions of Schubert polynomials for certain vexillary permutations. ...
Added: October 23, 2014
Arzhantsev I., Zaitseva Y., Russian Mathematical Surveys 2022 Vol. 77 No. 4 P. 571-650
We survey recent results on open embeddings of the affine space C^n into a complete algebraic variety X such that the action of the vector group G_a^n on C^n by translations extends to an action of G_a^n on X. We begin with the Hassett–Tschinkel correspondence describing equivariant embeddings of \mathbb{C}^n into projective spaces and present its generalization for embeddings into projective hypersurfaces. Further sections deal with embeddings into flag ...
Added: February 26, 2023
Kiritchenko V., Smirnov E., Timorin V., Успехи математических наук 2012 Т. 67 № 4 С. 89-128
We describe a new approach to the Schubert calculus on complete flag varieties using the volume polynomial associated with Gelfand-Zetlin polytopes. This approach allows us to compute the intersection products of Schubert cycles by intersecting faces of a polytope. ...
Added: September 19, 2012