Evgeny Feigin
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View article: Peter-Weyl Theorem for Iwahori Groups and Highest Weight Categories
Peter-Weyl Theorem for Iwahori Groups and Highest Weight Categories Open
We study the algebra of functions on the Iwahori group via the category of graded bounded representations of the Iwahori Lie algebra. In particular, we identify the standard and costandard objects in this category with certain generalized …
View article: Cauchy identities for staircase matrices
Cauchy identities for staircase matrices Open
The celebrated Cauchy identity expresses the product of terms $(1 - x_i y_j)^{-1}$ for $(i,j)$ indexing entries of a rectangular $m\times n$-matrix as a sum over partitions $λ$ of products of Schur polynomials: $s_λ(x)s_λ(y)$. Algebraicall…
View article: Birational maps, PBW degenerate flags and poset polytopes
Birational maps, PBW degenerate flags and poset polytopes Open
We extend the results on the graph closures of the birational maps between projective spaces and Grassmannians to the case of PBW degenerate flag varieties. The advantage of the PBW degenerate flags (as opposed to their classical analogues…
View article: Symplectic Grassmannians and Cyclic Quivers
Symplectic Grassmannians and Cyclic Quivers Open
The goal of this paper is to extend the quiver Grassmannian description of certain degenerations of Grassmann varieties to the symplectic case. We introduce a symplectic version of quiver Grassmannians studied in our previous papers and pr…
View article: Birational Maps to Grassmannians, Representations and Poset Polytopes
Birational Maps to Grassmannians, Representations and Poset Polytopes Open
We study the closure of the graph of the birational map from a projective space to a Grassmannian. We provide explicit description of the graph closure and compute the fibers of the natural projection to the Grassmannian. We construct embe…
View article: Birational maps to Grassmannians, representations and poset polytopes
Birational maps to Grassmannians, representations and poset polytopes Open
We study the closure of the graph of the birational map from a projective space to a Grassmannian. We provide explicit description of the graph closure and compute the fibers of the natural projection to the Grassmannian. We construct embe…
View article: Categorification of quiver diagonalization and Koszul algebras
Categorification of quiver diagonalization and Koszul algebras Open
In earlier work of three of the authors of the present paper, a supercommutative quadratic algebra was associated to each symmetric quiver, and a new proof of positivity of motivic Donaldson-Thomas invariants of symmetric quivers was given…
View article: PBW degenerations, quiver Grassmannians, and toric varieties
PBW degenerations, quiver Grassmannians, and toric varieties Open
We present a review on the recently discovered link between the Lie theory, the theory of quiver Grassmannians, and various degenerations of flag varieties. Our starting point is the induced Poincaré–Birkhoff–Witt filtration on the highest…
View article: Parahoric Lie algebras and parasymmetric Macdonald polynomials
Parahoric Lie algebras and parasymmetric Macdonald polynomials Open
The main goal of this paper is to categorify the specialized parasymmetric (intermediate) Macdonald polynomials. These polynomials depend on a parabolic subalgebra of a simple Lie algebra and generalize the symmetric and nonsymmetric Macdo…
View article: Peter-Weyl theorem for Iwahori groups and highest weight categories
Peter-Weyl theorem for Iwahori groups and highest weight categories Open
We study the algebra of functions on the Iwahori group via the category of graded bounded representations of its Lie algebra. In particular, we identify the standard and costandard objects in this category with certain generalized Weyl mod…
View article: Laumon parahoric local models via quiver Grassmannians
Laumon parahoric local models via quiver Grassmannians Open
Local models of Shimura varieties in type A can be realized inside products of Grassmannians via certain linear algebraic conditions. Laumon suggested a generalization which can be identified with a family over a line whose general fibers …
View article: Nonsymmetric $q$-Cauchy identity and representations of the Iwahori algebra
Nonsymmetric $q$-Cauchy identity and representations of the Iwahori algebra Open
The $t=0$ specialization of the Mimachi-Noumi Cauchy-type identity rewrites certain infinite product in terms of specialized nonsymmetric Macdonald polynomials of type $GL_n$. We interpret the infinite product as a character of the space o…
View article: Generalized juggling patterns, quiver Grassmannians and affine flag varieties
Generalized juggling patterns, quiver Grassmannians and affine flag varieties Open
The goal of this paper is to clarify the connection between certain structures from the theory of totally nonnegative Grassmannians, quiver Grassmannians for cyclic quivers and the theory of local models of Shimura varieties. More precisel…
View article: On reduced arc spaces of toric varieties
On reduced arc spaces of toric varieties Open
An arc space of an affine cone over a projective toric variety is known to be non-reduced in general. It was demonstrated recently that the reduced scheme structure is worth studying due to various connections with representation theory an…
View article: Reduced arc schemes for Veronese embeddings and global Demazure modules
Reduced arc schemes for Veronese embeddings and global Demazure modules Open
We consider arc spaces for the compositions of Plücker and Veronese embeddings of the flag varieties for simple Lie groups of types ADE. The arc spaces are not reduced and we consider the homogeneous coordinate rings of the corresponding r…
View article: Beyond the Sottile–Sturmfels Degeneration of a Semi-Infinite Grassmannian
Beyond the Sottile–Sturmfels Degeneration of a Semi-Infinite Grassmannian Open
We study toric degenerations of semi-infinite Grassmannians (a.k.a. quantum Grassmannians). While the toric degenerations of the classical Grassmannians are well studied, the only known example in the semi-infinite case is due to Sottile a…
View article: PBW degenerations, quiver Grassmannians, and toric varieties
PBW degenerations, quiver Grassmannians, and toric varieties Open
We present a review on the recently discovered link between the Lie theory,the theory of quiver Grassmannians, and various degenerations of flag varieties. Our starting point is the induced Poincaré--Birkhoff--Witt filtration on the highes…
View article: Koszul algebras and Donaldson-Thomas invariants
Koszul algebras and Donaldson-Thomas invariants Open
For a given symmetric quiver $Q$, we define a supercommutative quadratic\nalgebra $\\mathcal{A}_Q$ whose Poincar\\'e series is related to the motivic\ngenerating function of $Q$ by a simple change of variables. The Koszul duality\nbetween …
View article: Beyond the Sottile-Sturmfels degeneration of a semi-infinite\n Grassmannian
Beyond the Sottile-Sturmfels degeneration of a semi-infinite\n Grassmannian Open
We study toric degenerations of semi-infinite Grassmannians (a.k.a. quantum\nGrassmannians). While the toric degenerations of the classical Grassmannians\nare well studied, the only known example in the semi-infinite case is due to\nSottil…
View article: Totally nonnegative Grassmannians, Grassmann necklaces and quiver Grassmannians
Totally nonnegative Grassmannians, Grassmann necklaces and quiver Grassmannians Open
Postnikov constructed a cellular decomposition of the totally nonnegative Grassmannians. The poset of cells can be described (in particular) via Grassmann necklaces. We study certain quiver Grassmannians for the cyclic quiver admitting a c…
View article: Beilinson–Drinfeld Schubert varieties and global Demazure modules
Beilinson–Drinfeld Schubert varieties and global Demazure modules Open
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 …
View article: Semitoric degenerations of Hibi varieties and flag varieties
Semitoric degenerations of Hibi varieties and flag varieties Open
We construct a family of flat semitoric degenerations for the Hibi variety of every finite distributive lattice. The irreducible components of each degeneration are the toric varieties associated with polytopes forming a regular subdivisio…