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## Geometric mitosis

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, the algorithm yields a new combinatorial rule that extends to Sp_{2n}.

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

Kiritchenko V., Padalko M., / Cornell University. Series arXiv "math". 2018.

A Newton-Okounkov polytope of a complete flag variety can be turned into a convex geometric model for Schubert calculus. Namely, we can represent Schubert cycles by linear combinations of faces of the polytope so that the intersection product of cycles corresponds to the set-theoretic intersection of faces (whenever the latter are transverse). We explain the ...

Added: October 15, 2019

Kiritchenko V., Timorin V., Smirnov E., Oberwolfach Reports 2011 Vol. 8 No. 3 P. 2341-2344

We construct generalized Newton polytopes for Schubert subvarieties in the variety of complete flags in C^n . Every such “polytope” is a union of faces of a Gelfand–Zetlin polytope (the latter is a well-known Newton–Okounkov body for the flag variety). These unions of faces are responsible for Demazure characters of Schubert varieties and were originally used ...

Added: November 17, 2012

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

Kiritchenko V., Arnold Mathematical Journal 2019 Vol. 5 No. 2-3 P. 355-371

For classical groups SL(n), SO(n) and Sp(2n), we define uniformly geometric valuations on the corresponding complete flag varieties. The valuation in every type comes from a natural coordinate system on the open Schubert cell and is combinatorially related to the Gelfand-Zetlin pattern in the same type. In types A and C, we identify the corresponding ...

Added: October 15, 2019

Cherednik I., Feigin B. L., Advances in Mathematics 2013 Vol. 248 No. 25 P. 1050-1088

Using the DAHA-Fourier transform of q-Hermite polynomials multiplied by level-one theta functions, we obtain expansions of products of any number of such theta functions in terms of the q-Hermite polynomials. An ample family of modular functions satisfying Rogers-Ramanujan type identities for arbitrary (reduced, twisted) affine root systems is obtained as an application. A relation to ...

Added: September 30, 2013

Kiritchenko V., / Cornell University. Series arXiv "math". 2020.

A classical result of Schubert calculus is an inductive description of Schubert cycles using divided difference (or push-pull) operators in Chow rings. We define convex geometric analogs of push-pull operators and describe their applications to the theory of Newton-Okounkov convex bodies. Convex geometric push-pull operators yield an inductive construction of Newton-Okounkov polytopes of Bott-Samelson varieties. ...

Added: October 13, 2021

M. : Higher School of Economics Publishing House, 2012

Toric geometry exhibited a profound relation between algebra and topology on one side and combinatorics and convex geometry on the other side. In the last decades, the interplay between algebraic and convex geometry has been explored and used successfully in a much more general setting: first, for varieties with an algebraic group action (such as ...

Added: November 17, 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

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

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, , in : Oberwolfach Reports. Vol. 11. Issue 2.: Zürich : European Mathematical Society Publishing house, 2014. P. 1484-1487.

In [K], a convex-geometric algorithm was introduced for building new analogs of Gelfand–Zetlin polytopes for arbitrary reductive groups. Conjecturally, these polytopes coincide with the Newton–Okounkov polytopes of flag varieties for a geometric valuation. I outline an algorithm (geometric mitosis) for finding collec- tion of faces in these polytopes that represent a given Schubert cycle. For ...

Added: June 23, 2014

Mathematical Society of Japan, 2016

This volume contains the proceedings of the 5th MSJ Seasonal Institute on Schubert Calculus, held at Osaka City University, from September 17–27, 2012. It is recommended for all researchers and graduate students who are interested in Schubert calculus and its many connections and applications to related areas of mathematics, such as geometric representation theory, combinatorial ...

Added: October 19, 2020

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., 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

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

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

Краткое сообщение ...

Added: January 21, 2015

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

Kiritchenko V., Hornbostel J., Journal fuer die reine und angewandte Mathematik 2011 No. 656 P. 59-85

We establish a Schubert calculus for Bott-Samelson resolutions in the algebraic cobordism ring of a complete flag variety G/B. ...

Added: November 17, 2012

Evgeny Smirnov, Anna Tutubalina, European Journal of Combinatorics 2023 Vol. 107 Article 103613

Schubert polynomials for the classical groups were defined by S.Billey and M.Haiman in 1995; they are polynomial representatives of Schubert classes in a full flag variety over a classical group. We provide a combinatorial description for these polynomials, as well as their double versions, by introducing analogues of pipe dreams, or RC-graphs, for Weyl groups ...

Added: September 14, 2022

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

Kiritchenko V., Smirnov E., Timorin V., Russian Mathematical Surveys 2012 Vol. 67 No. 4 P. 685-719

A new approach is described to the Schubert calculus on complete flag varieties, using the volume polynomial associated with Gelfand- Zetlin polytopes. This approach makes it possible to compute the intersection products of Schubert cycles by intersecting faces of a polytope. Bibliography: 23 titles. ...

Added: February 4, 2013

Kiritchenko V., Квант 2014 № 1 С. 2-6

The article popularizes Schubert's method (Schubert calculus) for solving enumerative geometry problems. In particular, this method is applied to the classical problem on the number of lines that intersect 4 given lines in 3-space. The article in intended for high school students. ...

Added: May 16, 2014

Kiritchenko V., International Mathematics Research Notices 2023 Vol. 2023 No. 4 P. 3305-3328

A classical result of Schubert calculus is an inductive description of Schubert cycles using divided difference (or push-pull) operators in Chow rings. We define convex geometric analogs of push-pull operators and describe their applications to the theory of Newton-Okounkov convex bodies. Convex geometric push-pull operators yield an inductive construction of Newton-Okounkov polytopes of Bott-Samelson varieties. ...

Added: January 31, 2022