?
Gelfand-Zetlin polytopes and flag varieties
International Mathematics Research Notices. 2010. No. 13. P. 2512-2531.
I construct a correspondence between the Schubert cycles on the variety of complete flags in ℂn and some faces of the Gelfand–Zetlin polytope associated with the irreducible representation of SLn(ℂ) with a strictly dominant highest weight. The construction is motivated by the geometric presentation of Schubert cells using Demazure modules due to Bernstein–Gelfand–Gelfand [3]. The correspondence between the Schubert cycles and faces is then used to interpret the classical Chevalley formula in Schubert calculus in terms of the Gelfand–Zetlin polytopes. The whole picture resembles the picture for toric varieties and their polytopes.
Feigin E., Makedonskyi I., / Cornell University. Series math "arxiv.org". 2017. No. 1709.05674.
The goal of this paper is twofold. First, we write down the semi-infinite Pl\"ucker relations, describing the Drinfeld-Pl\"ucker embedding of the (formal version of) semi-infinite flag varieties in type A. Second, we study the homogeneous coordinate ring, i.e. the quotient by the ideal generated by the semi-infinite Pl\"ucker relations. We establish the isomorphism with the ...
Added: September 19, 2017
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
Dumanski, I., Feigin E., / Cornell University. Series math "arxiv.org". 2019. No. 1912.07988.
We consider the projective arc schemes of the Veronese embeddings of the flag
varieties for simple Lie groups of type ADE. The arc schemes are not reduced
and we consider the homogeneous coordinate rings of the corresponding reduced
schemes. We show that each graded component of a homogeneous coordinate ring is
a cocyclic module of the current algebra and ...
Added: December 18, 2019
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
Valentina Kiritchenko, / Cornell University. Series math "arxiv.org". 2013. No. 1307.7234.
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. This preprint contains the proofs of ...
Added: October 6, 2013
Kiritchenko Valentina, Timorin Vladlen, Gusev P., Journal of Combinatorial Theory, Series A 2013 Vol. 120 P. 960-969
We discuss the problem of counting vertices in Gelfand--Zetlin polytopes. Namely, we deduce a partial differential equation with constant coefficients on the exponential generating function for these numbers. For some particular classes of Gelfand-Zetlin polytopes, the number of vertices can be given by explicit formulas. ...
Added: February 18, 2013
Cerulli I., Fang X., Feigin E. et al., Mathematische Zeitschrift 2017 Vol. 287 No. 1 P. 615-654
Linear degenerate flag varieties are degenerations of flag varieties as quiver Grassmannians. For type A flag varieties, we obtain characterizations of flatness, irreducibility and normality of these degenerations via rank tuples. Some of them are shown to be isomorphic to Schubert varieties and can be realized as highest weight orbits of partially degenerate Lie algebras, ...
Added: February 17, 2017
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
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
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
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., 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
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
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, 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
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
Fang X., Feigin E., Fourier G. et al., Communications in Contemporary Mathematics 2019 Vol. 21 No. 1 P. 1-27
We study algebraic, combinatorial and geometric aspects of weighted Poincaré–Birkhoff–Witt (PBW)-type degenerations of (partial) flag varieties in type A. These degenerations are labeled by degree functions lying in an explicitly defined polyhedral cone, which can be identified with a maximal cone in the tropical flag variety. Varying the degree function in the cone, we recover, ...
Added: October 8, 2019
Feigin E., Makedonskyi I., Communications in Mathematical Physics 2019 Vol. 369 No. 1 P. 221-244
The direct sum of irreducible level one integrable representations of affine Kac-Moody Lie algebra of (affine) type ADE carries a structure of P/Q-graded vertex operator algebra. There exists a filtration on this direct sum studied by Kato and Loktev such that the corresponding graded vector space is a direct sum of global Weyl modules. The ...
Added: October 8, 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
Presnova E., Smirnov E., / Cornell University. Series math "arxiv.org". 2023. No. 2312.01417.
We give a new combinatorial description for stable Grothendieck polynomials in terms of subdivisions of Gelfand-Zetlin polytopes. Moreover, these subdivisions also provide a description of Lascoux polynomials. This generalizes a similar result on key polynomials by Kiritchenko, Smirnov, and Timorin. ...
Added: February 12, 2024
Kalashnikov E. G., / arXiv. Series arXiv "arXiv". 2020.
We introduce a superpotential for partial flag varieties of type A. This is a map W:Y∘→C, where Y∘ is the complement of an anticanonical divisor on a product of Grassmannians. The map W is expressed in terms of Plücker coordinates of the Grassmannian factors. This construction generalizes the Marsh--Rietsch Plücker coordinate mirror for Grassmannians. We show that in a distinguished cluster ...
Added: November 26, 2020
Deviatov R., / Cornell University. Series math "arxiv.org". 2010. No. arXiv:1007.1353v1.
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 [7], we enumerate all triples (G, P, n) such that (a) there exists an open G-orbit on the multiple flag variety ...
Added: June 28, 2012
Makhlin I., Selecta Mathematica, New Series 2015
We exploit the idea that the character of an irreducible finite dimensional $\mathfrak{gl}_n$-module is the sum of certain exponents of integer points in a Gelfand-Tsetlin polytope and can thus be calculated via Brion's theorem. In order to show how the result of such a calculation matches Weyl's character formula we prove some interesting combinatorial traits ...
Added: September 29, 2014
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