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## Degenerate affine Grassmannians and loop quivers

Cornell University
,
2014.
No. 1410.0777.

We study the connection between the affine degenerate Grassmannians in type $A$, quiver Grassmannians for one vertex loop quivers and affine
Schubert varieties. We give an explicit description of the degenerate affine Grassmannian of type $GL_n$ and identify it
with semi-infinite orbit closure of type $A_{2n-1}$. We show that principal quiver Grassmannians for the one vertex loop quiver
provide finite-dimensional approximations of the degenerate affine Grassmannian. Finally, we give an explicit
description of the degenerate affine Grassmannian of type $A_1^{(1)}$, propose
a conjectural description in the symplectic case and discuss the generalization to the case of the affine degenerate flag varieties.

Shirokov D., Advances in Applied Clifford Algebras 2015 Vol. 25 No. 3 P. 707-718

In this paper we prove isomorphisms between 5 Lie groups (of arbitrary dimension and fixed signatures) in Clifford algebra and classical matrix Lie groups - symplectic, orthogonal and linear groups. Also we obtain isomorphisms of corresponding Lie algebras. ...

Added: March 12, 2015

Feigin E., Functional Analysis and Its Applications 2014 Vol. 48 No. 1 P. 59-71

The degenerate Lie group is a semidirect product of the Borel subgroup with the normal
abelian unipotent subgroup.
We introduce a class of the highest weight representations of the degenerate group of type A, generalizing
the PBW-graded representations of the classical group. Following the classical construction
of the flag varieties, we consider the closures of the orbits of the ...

Added: April 30, 2014

Gritsenko V., Ванг Х., Успехи математических наук 2017 Т. 72 № 5 С. 191-192

In this paper we prove the conjecture above in the last case of known theta-blocks of weight 2. This gives a new intereting series of Borcherds products of weight 2. ...

Added: October 11, 2017

Feigin E., Makedonskyi I., / Cornell University. Series math "arxiv.org". 2015. No. 1512.03254.

The classical local Weyl modules for a simple Lie algebra are labeled by dominant weights. We generalize the definition to the case of arbitrary weights and study the properties of the generalized modules. We prove that the representation theory of the generalized Weyl modules can be described in terms of the alcove paths and the ...

Added: December 15, 2015

Feigin B. L., Функциональный анализ и его приложения 2014 № 3

We study commutative vertex operator algebras. These algebras are isomorphic to certain subalgebras in Kac-Moody vertex operator algebras. We describe systems of relations and degenerations to quadratic algebras. Our approach leads to the fermionic formulas for characters. ...

Added: April 14, 2014

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., 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 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

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

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

Smirnov E., Bulletin of the Korean Mathematical Society 2017 Vol. 54 No. 5 P. 1773-1778

We give an alternative proof of a recent result by Pasquier stating that for a generalized flag variety X=G/P and an effective Q-divisor D stable with respect to a Borel subgroup the pair (X,D) is Kawamata log terminal if and only if [D]=0. ...

Added: February 14, 2017

Arzhantsev I., Makedonskii E. A., Petravchuk A. P., Украинский математический журнал 2011 Vol. 63 No. 5 P. 708-712

Added: July 10, 2014

Roman Avdeev, Petukhov A., Transformation Groups 2021 Vol. 26 No. 3 P. 719-774

Let G be a symplectic or special orthogonal group, let H be a connected reductive subgroup of G, and let X be a flag variety of G. We classify all triples (G, H, X) such that the natural action of H on X is spherical. For each of these triples, we determine the restrictions to ...

Added: September 2, 2020

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

Р.С. Авдеев, Петухов А. В., Математический сборник 2014 Т. 205 № 9 С. 3-48

For every finite-dimensional vector space V and every V-flag variety X we list all connected reductive subgroups in GL(V) acting spherically on X. ...

Added: October 22, 2014

Makedonskyi I., Петравчук А. П., Journal of Algebra 2014 Vol. 401 P. 245-257

Let K be a ﬁeld and A be a commutative associative K-algebra which is an integral domain. The Lie algebra DerA of all K-derivations of A is an A-module in a natural way and if R is the quotient ﬁeld of A then RDerA is a vector space over R. It is proved that if ...

Added: March 20, 2014

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

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

Merzon G., Smirnov E., / Cornell University. Series math "arxiv.org". 2014. No. 1410.6857.

We prove new determinantal identities for a family of flagged Schur polynomials. As a corollary of these identities we obtain determinantal expressions of Schubert polynomials for certain vexillary permutations. ...

Added: October 23, 2014

Gritsenko V., Успехи математических наук 2018 Т. 73 № 5 С. 53-122

Рефлективные модулярные формы ортогонального типа — это фундаментальные автоморфные объекты, обобщающие классическую эта-функцию Дедекинда. В этой статье мы опишем в терминах форм Якоби две конструкции для построения таких модулярных форм: автоморфные произведения и подъем Якоби. В частности, мы докажем, что первый коэффициент Фурье--Якоби модулярной формы Борчердса $\Phi_{12}$
(производящая функция для ``Fake Monster Lie Algebra’’) в любом из 23 ...

Added: October 2, 2018

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

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