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## Слайд-многочлены и комплексы подслов

С. 43-44.

We define simplicial complexes for slide polynomials and show that they are always homeomorphic to balls or spheres.

### In book

Самара: Инсома-пресс, 2018

Kiritchenko V., Maria Padalko, Schubert calculus on Newton-Okounkov polytopes / 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

Degtiarev Konstantin Y., , in: Proceedings of the 8-th International Conference on Soft Computing, Computing with Words and Perceptions in System Analysis, Decision and Control (ICSCCW-2015). .: Kaufering: b-Quadrat Verlag, 2015.. P. 45-55.

In this paper we propose probability-theoretic approach to compare intervals with rigorous bounds which opens a door to more adequate forms of representation and processing of expert estimates uncertainty ...

Added: February 28, 2016

Kiritchenko V., , in: Oberwolfach Reports. Issue 9.: Oberwolfach: European Mathematical Society Publishing house, 2012.. P. 5-7.

I describe a convex geometric procedure for building generalized Newton polytopes of Schubert varieties. One of the goals is to extend to arbitrary reductive groups our joint work with Evgeny Smirnov and Vladlen Timorin on Schubert calculus (in type A) in terms of Gelfand-Zetlin polytopes. ...

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

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, 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

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

Kiritchenko V., Jens Hornbostel, 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

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

Degtyarev K. Y., , in: Proceedings of the 10th International Conference on Application of Fuzzy Systems and Soft Computing (ICAFS-2012). .: Kaufering: b-Quadrat Verlag, 2012.. P. 27-37.

The material of the present paper is grounded on the holist algebraic method (Q-analysis) proposed by English mathematician and physicist R.H.Atkin. At its core, the approach is aimed at both analysis of systems structures (in the form of simplicial complexes K, which is formed by a set of properly adjoined objects called simplexes) and calculation ...

Added: October 31, 2012

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

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

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

Smirnov E., Тутубалина А. А., Успехи математических наук 2020 Т. 75 № 6(456) С. 177-178

В работе рассматривается подразбиение комплексов подслов, определенных Кнутсоном и Миллером, на слайд-комплексы; показано, что эти комплексы являются шеллинговыми и гомеоморфны шару или сфере. ...

Added: October 28, 2020

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

Smirnov E., Тутубалина А. А., Slide polynomials and subword complexes / Cornell University. Series math "arxiv.org". 2020. No. 2006.16995.

Subword complexes were defined by A.Knutson and E.Miller in 2004 for describing Gröbner degenerations of matrix Schubert varieties. The facets of such a complex are indexed by pipe dreams, or, equivalently, by the monomials in the corresponding Schubert polynomial. In 2017 S.Assaf and D.Searles defined a basis of slide polynomials, generalizing Stanley symmetric functions, and ...

Added: July 6, 2020