arXiv subjects

Evgeny Feigin

Publications and source records attributed to Evgeny Feigin.

At least 19 recordsLinked to original sources

arXiv2026

Extremality of principal quiver Grassmannians

We study families of quiver Grassmannians, parametrizing subrepresentations of representations of Dyn\-kin quivers, which are naturally associated to a projective and an injective representation. Our main result is an explicit description of the flat irreducible locus of such a family. More precisely, we prove that a member of this family is irreducible and of the expected dimension if and only if the ambient quiver representation degenerates to the direct sum of the defining projective and injective representations.

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arXiv2026

Cauchy identities for staircase matrices

The well known 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: $s_λ(x)s_λ(y)$. Algebraically, this identity comes from the decomposition of the symmetric algebra of the space of rectangular matrices, considered as a $\mathfrak{gl}_m$-$\mathfrak{gl}_n$-bimodule. We generalize the Cauchy decomposition by replacing rectangular matrices with arbitrary staircase-shaped matrices equipped with the left and right actions of the Borel upper-triangular subalgebras. For any given staircase shape $\mathsf{Y}$ we describe left and right ``standard" filtrations on the symmetric algebra of the space of shape $\mathsf{Y}$ matrices. We show that the subquotients of these filtrations are tensor products of Demazure and opposite van der Kallen modules over the Borel subalgebras. On the level of characters, we derive two distinct expansions for the product $(1 - x_i y_j)^{-1}$ for $(i,j) \in \mathsf{Y}$ written as sums of products of key polynomials $κ_λ(x)$ and (opposite) Demazure atoms $a^μ(y)$.

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arXiv2026

Type A algebraic coherence conjecture of Pappas and Rapoport

The Pappas--Rapoport coherence conjecture, proved by Zhu, states that the dimensions of spaces of sections of certain line bundles coincide. The two sides of the equality correspond to line bundles on spherical Schubert varieties in affine Grassmannians and to line bundles on unions of Schubert varieties in affine flag varieties. Algebraically, the claim can be reformulated as an equality between the dimensions of certain Demazure modules and certain sums of Demazure modules. The goal of this paper is to formulate an algebraic construction that provides an explicit link between the aforementioned Demazure modules. Our construction works only in type A, but it applies to a much wider class of representations than those arising in the geometric coherence conjecture. In the general case, one side of the conjectural equality involves affine Kostant--Kumar modules.

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arXiv2026

Truncated Grassmannians, blow-ups along Schubert varieties and collineations

Truncated Grassmannians are defined as closures of orbits of abelian unipotent groups acting on the degree truncations of projectivized wedge powers. We show that such truncations in a more general setup show up in the description of the blow-ups of general flag varieties along Schubert subvarieties. We work out the case of Grassmannians in detail. In particular, we show that our blow-ups are members of a larger family of varieties projecting onto Grassmannians, and describe the fibers of these projections via the spaces of collineations.

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arXiv2025

PrIncipal quiver Grassmannians: conjectures

Let $P$ and $I$ be a projective and an injective representations of a Dynkin quiver. We consider quiver Grassmannians of subrepresentations of dimension $\dim P$ inside representations of dimension $\dim P + \dim I$. Based on extensive computer experiments, we formulate several conjectures about the algebro-geometric properties of these quiver Grassmannians.

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arXiv2025

From brick manifolds to Grassmannians of bimodules

We study a class of Grassmannians of sub-bimodules over the path algebras of quivers. Our quiver Grassmannians include Escobar's brick manifolds as well as Labelle's generalizations. We give an explicit construction of the varieties in question, provide examples and clarify connection with the quiver representation spaces. We also prove smoothness of our Grassmannians, construct cellular decompositions and derive a realization as framed moduli spaces. The framed moduli realization leads to a recursive formula for the motives.

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arXiv2025

Global positroid varieties

Positroid subvarieties of complex Grassmannians are the images of the Richardson subvarieties of the full flag varieties under the natural projection map. Positroid varieties admit natural embedding into certain quiver Grassmannians for equioriented cyclic quivers. Varying representations of the quiver, one defines global positroid varieties inside the type A global affine Grassmannian. General fiber of a family is isomorphic to the corresponding classical positroid variety and the special fiber is a subvariety of the juggling variety. We show that the global positroid families are flat and describe their (conjecturally reduced) scheme structures. We also describe the special fibers inside the product of classical positroid varieties and realize the irreducible components of the special fibers in terms of the affine Richardson varieties.

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arXiv2025

Peter-Weyl theorem for Iwahori groups and highest weight categories

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 modules. Using this identification we express the characters of the standard and costandard objects in terms of specialized nonsymmetric Macdonald polynomials. We also prove that our category of interest admits a generalized highest weight structure (known as stratified structure). We show, more generally, that such a structure on a category of representations of a Lie algebra implies the Peter-Weyl type theorem for the corresponding algebraic group. In the Iwahori case, standard filtrations of indecomposable projective objects correspond to new ``reciprocal'' Macdonald-type identities.

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arXiv2025

On reduced arc spaces of toric varieties

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 and combinatorics. In this paper we develop a general machinery for the description of the reduced arc spaces of affine cones over toric varieties. We apply our techniques to a number of classical cases and explore some connections with representation theory of current algebras.

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arXiv2024

Symplectic Grassmannians and Cyclic Quivers

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 prove a number of results on these projective algebraic varieties. First, we construct a cellular decomposition of the symplectic quiver Grassmannians in question and develop combinatorics needed to compute Euler characteristics and Poincaré polynomials. Second, we show that the number of irreducible components of our varieties coincides with the Euler characteristic of the classical symplectic Grassmannians. Third, we describe the automorphism groups of the underlying symplectic quiver representations and show that the cells are the orbits of this group. Lastly, we provide an embedding into the affine flag varieties for the affine symplectic group.

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arXiv2024

Birational maps, PBW degenerate flags and poset polytopes

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) is the existence of a large group of symmetries for the graph closures. We discuss the combinatorial, algebraic and geometric sides of the picture. In particular, we show that toric degenerations of Borovik, Sturmfels and Sverrisdóttir are still available in the general settings. We also derive a description of the graph closures for flag varieties in terms of quiver representations.

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arXiv2024

Birational maps to Grassmannians, representations and poset polytopes

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 embeddings of the graph closure to the projectivizations of certain cyclic representations of a degenerate special linear Lie algebra and study algebraic and combinatorial properties of these representations. In particular, we describe monomial bases, generalizing the FFLV bases. The proof relies on combinatorial properties of a new family of poset polytopes, which are of independent interest. As a consequence we obtain flat toric degenerations of the graph closure studied by Borovik, Sturmfels and Sverrisdóttir.

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arXiv2024

Relative poset polytopes and semitoric degenerations

The two best studied toric degenerations of the flag variety are those given by the Gelfand--Tsetlin and FFLV polytopes. Each of them degenerates further into a particular monomial variety which raises the problem of describing the degenerations intermediate between the toric and the monomial ones. Using a theorem of Zhu one may show that every such degeneration is semitoric with irreducible components given by a regular subdivision of the corresponding polytope. This leads one to study the parts that appear in such subdivisions as well as the associated toric varieties. It turns out that these parts lie in a certain new family of poset polytopes which we term relative poset polytopes: each is given by a poset and a weakening of its order relation. In this paper we give an in depth study of (both common and marked) relative poset polytopes and their toric varieties in the generality of an arbitrary poset. We then apply these results to degenerations of flag varieties. We also show that our family of polytopes generalizes the family studied in a series of papers by Fang, Fourier, Litza and Pegel while sharing their key combinatorial properties such as pairwise Ehrhart-equivalence and Minkowski-additivity.

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arXiv2024

Categorification of quiver diagonalization and Koszul algebras

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 using the so called numerical Koszul property of these algebras. It was furthermore conjectured that for each symmetric quiver such an algebra is Koszul. In this work, we lift the linking and unlinking operations on symmetric quivers of Ekholm, Longhi and the third author to the level of quadratic algebras, and use those lifts to prove the Koszulness conjecture.

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arXiv2023

Parahoric Lie algebras and parasymmetric Macdonald polynomials

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 Macdonald polynomials. To achieve this we introduce cyclic modules of the parahoric subalgebras of the affine Kac-Moody Lie algebras such that their characters coincide with the specializations of the parasymmetric polynomials at zero and infinity. These cyclic modules are proved to coincide with standard and costandard objects in certain categories of representations of parahoric algebras. We show that the categories in question are stratified, i.e. they are graded highest weight categories. As a consequence, we derive an analog of the Peter-Weyl theorem describing the bi-module of functions on the parahoric and parabolic Lie groups via the mentioned above standard and costandard modules.

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arXiv2023

Laumon parahoric local models via quiver Grassmannians

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 are quiver Grassmannians for the loop quiver and the special fiber is a certain quiver Grassmannian for the cyclic quiver. The whole family sits inside the Gaitsgory central degeneration of the affine Grassmannians. We study the properties of the special fibers of the (complex) Laumon local models for arbitrary parahoric subgroups in type A using the machinery of quiver representations. We describe the irreducible components and the natural strata with respect to the group action for the quiver Grassmannians in question. We also construct a cellular decomposition and provide an explicit description for the corresponding poset of cells. Finally, we study the properties of the desingularizations of the irreducible components and show that the desingularization construction is compatible with the natural projections between the parahoric subgroups.

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arXiv2023

Nonsymmetric $q$-Cauchy identity and representations of the Iwahori algebra

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 of functions on a certain matrix space. We show that the space of functions admits a filtration such that the graded pieces are isomorphic to the tensor products of certain generalized global Weyl modules of the Iwahori algebra. We identify the characters of the graded pieces with the terms of the specialized Mimachi-Noumi formula. We conjecture the existence of an analogous filtration on the space of functions on the Iwahori group for all simple Lie algebras and prove the conjecture for $SL_n$. Our construction can be seen as a current algebra extension of the van der Kallen filtration on functions on a Borel subgroup.

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arXiv2023

Generalized juggling patterns, quiver Grassmannians and affine flag varieties

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 precisely, we generalize the construction from our previous paper relating the combinatorics and geometry of quiver Grassmanians to that of the totally nonnegative Grassmannians. The varieties we are interested in serve as realizations of local models of Shimura varieties. We exploit quiver representation techniques to study the quiver Grassmannians of interest and, in particular, to describe explicitly embeddings into affine flag varieties which allow us to realize our quiver Grassmannians as a union of Schubert varieties therein.

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