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## Divided difference operators on polytopes

Ch. 6. P. 161-185.

We define convex-geometric counterparts of divided difference (or Demazure) operators from the Schubert calculus and representation theory. These operators are used to construct inductively polytopes that capture Demazure characters of representations of reductive groups. In particular, Gelfand-Zetlin polytopes and twisted cubes of Grossberg-Karshon are obtained in a uniform way.

Keywords: Newton-Okounkov polytopesмногогранники Ньютона-ОкуньковаDemazure operatorоператоры Демазюра

Publication based on the results of:

### In book

Mathematical Society of Japan, 2016

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, 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

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., 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

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

Kiritchenko V., Padalko M., 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

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