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## Schubert calculus on Newton-Okounkov polytopes

math.
arXiv.
Cornell University
,
2018.

Kiritchenko V., Padalko M.

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 general framework and survey particular realizations of this approach in types A, B and C.

Kiritchenko V., Geometric mitosis / 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

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

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

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

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., Push-pull operators on convex polytopes / 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

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

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

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

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

Valentina Kiritchenko, Newton-Okounkov polytopes of Bott-Samelson varieties as Minkowski sums / 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., 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

Kiritchenko V., International Mathematics Research Notices 2021

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

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

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

Borzykh D., ЛЕНАНД, 2021

Книга представляет собой экспресс-курс по теории вероятностей в контексте начального курса эконометрики. В курсе в максимально доступной форме изложен тот минимум, который необходим для осознанного изучения начального курса эконометрики. Данная книга может не только помочь ликвидировать пробелы в знаниях по теории вероятностей, но и позволить в первом приближении выучить предмет «с нуля». При этом, благодаря доступности изложения и небольшому объему книги, ...

Added: February 20, 2021

Kotelnikova M. V., Aistov A., Вестник Нижегородского университета им. Н.И. Лобачевского. Серия: Социальные науки 2019 Т. 55 № 3 С. 183-189

The article describes a method that allows to improve the content of disciplines of the mathematical cycle by dividing them into invariant (general) and variable parts. The invariants were identified for such disciplines as «Linear algebra», «Mathematical analysis», «Probability theory and mathematical statistics» delivered to Bachelors program students of economics at several universities. Based on ...

Added: January 28, 2020

В. Л. Попов, Математические заметки 2017 Т. 102 № 1 С. 72-80

Мы доказываем, что аффинно-треугольные подгруппы являются борелевскими подгруппами групп Кремоны. ...

Added: May 3, 2017

Красноярск : ИВМ СО РАН, 2013

Труды Пятой Международной конференции «Системный анализ и информационные технологии» САИТ-2013 (19–25 сентября 2013 г., г.Красноярск, Россия): ...

Added: November 18, 2013

Min Namkung, Younghun K., Scientific Reports 2018 Vol. 8 No. 1 P. 16915-1-16915-18

Sequential state discrimination is a strategy for quantum state discrimination of a sender’s quantum
states when N receivers are separately located. In this report, we propose optical designs that can
perform sequential state discrimination of two coherent states. For this purpose, we consider not
only binary phase-shifting-key (BPSK) signals but also general coherent states, with arbitrary prior
probabilities. Since ...

Added: November 16, 2020

Grines V., Gurevich E., Pochinka O., Russian Mathematical Surveys 2017 Vol. 71 No. 6 P. 1146-1148

In the paper a Palis problem on finding sufficient conditions on embedding of Morse-Smale diffeomorphisms in topological flow is discussed. ...

Added: May 17, 2017

Okounkov A., Aganagic M., Moscow Mathematical Journal 2017 Vol. 17 No. 4 P. 565-600

We associate an explicit equivalent descendent insertion to any relative insertion in quantum K-theory of Nakajima varieties.
This also serves as an explicit formula for off-shell Bethe eigenfunctions for general quantum loop algebras associated to quivers and gives the general integral solution to the corresponding quantum Knizhnik Zamolodchikov and dynamical q-difference equations. ...

Added: October 25, 2018

Danilov B.R., Moscow University Computational Mathematics and Cybernetics 2013 Vol. 37 No. 4 P. 180-188

The article investigates a model of delays in a network of functional elements (a gate network) in an arbitrary finite complete basis B, where basis elements delays are arbitrary positive real numbers that are specified for each input and each set of boolean variables supplied on the other inputs. Asymptotic bounds of the form τ ...

Added: December 2, 2019