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## Multiple flag varieties

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/P*1*×· · ·×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 number of *G*-orbits and existence of an open *G*-orbits on a multiple flag variety.

Publication based on the results of:

Kiritchenko Valentina, Krishna A., Transformation Groups 2013 Vol. 18 No. 2 P. 391-413

We obtain an explicit presentation for the equivariant cobordism ring of a complete flag variety. An immediate corollary is a Borel presentation for the ordinary cobordism ring. Another application is an equivariant Schubert calculus in cobordism. We also describe the rational equivariant cobordism rings of wonderful symmetric varieties of minimal rank. ...

Added: February 18, 2013

Bigeni A., Feigin E., Linear Algebra and its Applications 2019 Vol. 573 P. 54-79

The goal of this paper is to study the link between the topology of the degenerate flag varieties and combinatorics of the Dellac configurations. We define three new classes of algebraic varieties closely related to the degenerate flag varieties of types A and C. The definitions are given in terms of linear algebra: they are ...

Added: October 8, 2019

Cerulli Irelli G., Fang X., Feigin E. et al., / Cornell University. Series math "arxiv.org". 2019. No. 1901.11020.

We continue, generalize and expand our study of linear degenerations of flag varieties from [G. Cerulli Irelli, X. Fang, E. Feigin, G. Fourier, M. Reineke, Math. Z. 287 (2017), no. 1-2, 615-654]. We realize partial flag varieties as quiver Grassmannians for equi-oriented type A quivers and construct linear degenerations by varying the corresponding quiver representation. ...

Added: February 5, 2019

Bigeni A., Feigin E., Journal of Integer Sequences 2020 Vol. 23 No. 20.4.6 P. 1-32

We define symmetric Dellac configurations as the Dellac configurations that are symmetrical with respect to their centers. The even-length symmetric Dellac configurations coincide with the Fang-Fourier symplectic Dellac configurations. Symmetric Dellac configurations generate the Poincaré polynomials of (odd or even) symplectic or orthogonal versions of degenerate flag varieties. We give several combinatorial interpretations of the ...

Added: April 16, 2020

Feigin E., Fourier G., Littelmann P., Transformation Groups 2017 Vol. 22 No. 2 P. 321-352

We introduce the notion of a favourable module for a complex unipotent algebraic group, whose properties are governed by the combinatorics of an associated polytope. We describe two filtrations of the module, one given by the total degree on the PBW basis of the corresponding Lie algebra, the other by fixing a homogeneous monomial order ...

Added: August 4, 2017

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

Arzhantsev I., Proceedings of the American Mathematical Society 2011 Vol. 139 No. 3 P. 783-786

Added: July 10, 2014

Tyurin N. A., Математические заметки 2014 Т. 96 № 3 С. 476-479

Краткое сообщение ...

Added: January 21, 2015

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

Altmann K., Kiritchenko V., Petersen L., Michigan Mathematical Journal 2015 Vol. 64 P. 3-38

Given a spherical homogeneous space G/H of minimal rank, we provide a simple procedure to describe its embeddings as varieties with torus action in terms of divisorial fans. The torus in question is obtained as the identity component of the quotient group N/H, where N is the normalizer of H in G. The resulting Chow ...

Added: April 3, 2015

Cerulli Irelli G., Fang X., Feigin E. et al., Mathematische Zeitschrift 2020 Vol. 296 No. 1 P. 453-477

We continue, generalize and expand our study of linear degenerations of flag varieties from Cerulli Irelli et al. (Math Z 287(1–2):615–654, 2017). We realize partial flag varieties as quiver Grassmannians for equi-oriented type A quivers and construct linear degenerations by varying the corresponding quiver representation. We prove that there exists the deepest flat degeneration and the ...

Added: September 1, 2020

Kiritchenko V., Annales de l'Institut Fourier 2006 Vol. 56 No. 4 P. 1225-1256

In this paper, I construct noncompact analogs of the Chern classes of equivariant vector bundles over complex reductive groups. For the tangent bundle, these Chern classes yield an adjunction formula for the Euler characteristic of complete intersections in reductive groups. When a complete intersection is a curve, this formula gives an explicit answer for the ...

Added: October 7, 2013

Zhgoon V., Knop F., Doklady Mathematics 2019 Vol. 99 No. 2 P. 132-136

We prove new results that generalize Vinberg’s complexity theorem for the action of reductive group on an algebraic variety over an algebraically nonclosed field. We provide new results on strong k-stability for actions on flag varieties are given. ...

Added: October 8, 2019

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

Valentina Kiritchenko, Transformation Groups 2017 Vol. 22 No. 2 P. 387-402

We compute the Newton-Okounkov bodies of line bundles on the complete flag variety of GL_n for a geometric valuation coming from a flag of translated Schubert subvarieties. The Schubert subvarieties correspond to the terminal subwords in the decomposition (s_1)(s_2s_1)(s_3s_2s_1)(...)(s_{n-1}...s_1) of the longest element in the Weyl group. The resulting Newton-Okounkov bodies coincide with the Feigin-Fourier-Littelmann-Vinberg ...

Added: February 25, 2016

Aleksei Golota, / Cornell University. Series arXiv "math". 2019.

For a polarized variety (X,L) and a closed connected subgroup G⊂Aut(X,L) we define a G-invariant version of the δ-threshold. We prove that for a Fano variety (X,−KX) and a connected subgroup G⊂Aut(X) this invariant characterizes G-equivariant uniform K-stability. We also use this invariant to investigate G-equivariant K-stability of some Fano varieties with large groups of ...

Added: October 7, 2019

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

Valentina Kiritchenko, / Cornell University. Series arXiv "math". 2018.

We compute the Newton--Okounkov bodies of line bundles on a Bott--Samelson resolution of the complete flag variety of $GL_n$ for a geometric valuation coming from a flag of translated Schubert subvarieties. The Bott--Samelson resolution corresponds to the decomposition (s_1)(s_2s_1)(s_3s_2s_1)(...)(s_{n-1}\ldots s_1) of the longest element in the Weyl group, and the Schubert subvarieties correspond to the ...

Added: August 20, 2018

Smirnov E., В кн. : Труды семинара по алгебре и геометрии Самарского университета. Т. 147.: М. : ВИНИТИ РАН, 2018. Гл. 3. С. 84-119.

Работа посвящена обзору основных результатов о кратных многообразиях флагов. Приведена классификация кратных многообразий флагов сложности 0 и 1 и изложены результаты о комбинаторике и геометрии B-орбит и их замыканий в двойных комикровесовых многообразиях флагов. Также обсуждаются вопросы конечности числа G-орбит на кратном многообразии флагов и существования на нем открытой G-орбиты. ...

Added: August 19, 2018

Feigin E., Selecta Mathematica, New Series 2012 Vol. 18 No. 3 P. 513-537

Let Fλ be a generalized flag variety of a simple Lie group G embedded into the projectivization of an irreducible G-module Vλ. We define a flat degeneration Fλa, which is a GaM variety. Moreover, there exists a larger group Ga acting on Fλa, which is a degeneration of the group G. The group Ga contains ...

Added: August 31, 2012

Feigin E., Cerulli Irelli G., Reineke M., Algebra & Number Theory 2012 Vol. 6 No. 1 P. 165-194

Quiver Grassmannians are varieties parametrizing subrepresentations of a quiver representation. It is observed that certain quiver Grassmannians for type A quivers are isomorphic to the degenerate flag varieties investigated earlier by the second named author. This leads to the consideration of a class of Grassmannians of subrepresentations of the direct sum of a projective and ...

Added: June 29, 2012

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

Rostislav Devyatov, International Mathematics Research Notices 2014 Vol. 2014 No. 11 P. 2972-2989

Let G be a semisimple algebraic group whose decomposition into the product of simple components does not contain simple groups of type A, and P⊆G be a parabolic subgroup. Extending the results of Popov, we enumerate all triples (G,P,n) such that (a) there exists an open G-orbit on the multiple flag variety G/P×G/P×⋯×G/P (n factors) ...

Added: October 9, 2013

Feigin E., Makedonskyi I., International Mathematics Research Notices 2020 No. 14 P. 4357-4394

The goal of this paper is two-fold. First, we write down the semi-infinite Plücker relations, describing the Drinfeld–Plücker embedding of the (formal version of) semi-infinite flag varieties in type A. Second, we study the homogeneous coordinate ring, that is, the quotient by the ideal generated by the semi-infinite Plücker relations. We establish the isomorphism with ...

Added: September 1, 2020