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## Reduced arc schemes for Veronese embeddings and global Demazure modules

Cornell University
,
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 is acted upon by the algebra of
symmetric polynomials. We show that the action of the polynomial algebra is
free and that the localization at the special point of a graded component is
isomorphic to an affine Demazure module whose level is the degree of the
Veronese embedding. In type $A_1$ we give the precise list of generators of the
reduced arc scheme structure of the Veronese curves. In general type we
introduce the notion of the global higher level Demazure modules and identify
the graded components of the homogeneous coordinate rings with these modules.

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

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

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

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

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

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

Ilya Dumanski, Evgeny Feigin, Michael Finkelberg, Forum of Mathematics, Sigma 2021 Vol. 9 P. 1-25

We compute the spaces of sections of powers of the determinant line bundle on the spherical Schubert subvarieties
of the Beilinson–Drinfeld aﬃne Grassmannians. The answer is given in terms of global Demazure modules over
the current Lie algebra. ...

Added: May 25, 2021

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

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

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

Kiritchenko V., 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 ...

Added: November 17, 2012

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

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

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

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

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

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

Netay I. V., Математический сборник 2017 Т. 208 № 2 С. 31-54

Явно вычислены сизигии квадратичного вложения Веронезе P(V)⊂P(Sym2V) как представления группы GL(V). Также построены резольвенты для пучков OP(V)(i) в категории D(P(Sym2V)). ...

Added: April 10, 2017

Cherednik I., Feigin E., Advances in Mathematics 2015 Vol. 282 P. 220-264

Given a reduced irreducible root system, the corresponding nil-DAHA is used to calculate the extremal coefficients of nonsymmetric Macdonald polynomials in the limit t→∞ and for antidominant weights, which is an important ingredient of the new theory of nonsymmetric q-Whittaker function. These coefficients are pure q-powers and their degrees are expected to coincide in the ...

Added: September 3, 2015

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

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