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Evgeny Feigin

Publications and source records attributed to Evgeny Feigin.

At least 37 records · Page 2Linked to original sources
arXiv2022

Beyond the Sottile-Sturmfels degeneration of a semi-infinite Grassmannian

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-Sturmfels. We start by providing a new interpretation of the Sottile-Sturmfels construction by finding a poset such that their degeneration is the toric variety of the order polytope of the poset. We then use our poset to construct and study a new toric degeneration in the semi-infinite case. Our construction is based on the notion of poset polytopes introduced by Fang-Fourier-Litza-Pegel. As an application we introduce semi-infinite PBW-semistandard tableaux, giving a basis in the homogeneous coordinate ring of a semi-infinite Grassmannian.

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arXiv2022

Reduced arc schemes for Veronese embeddings and global Demazure modules

We consider arc spaces for the compositions of Pluecker 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 reduced schemes. We show that each graded component of a homogeneous coordinate ring is a cocyclic module over 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 fiber 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$_1$ (which corresponds to the Veronese curve) we give the precise list of generators of the reduced arc space. In general type, we introduce the notion of global higher level Demazure modules, which generalizes the standard notion of the global Weyl modules, and identify the graded components of the homogeneous coordinate rings with these modules.

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arXiv2022

PBW degenerations, quiver Grassmannians, and toric varieties

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 weight representations and the corresponding PBW degenerate flag varieties.

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arXiv2021

Semitoric degenerations of Hibi varieties and flag varieties

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 subdivision of the order polytope of the underlying poset. These components are themselves Hibi varieties. For each degeneration in our family we also define the corresponding weight polytope and embed the degeneration into the associated toric variety as the union of orbit closures given by a set of faces. Every such weight polytope projects onto the order polytope with the chosen faces projecting into the parts of the corresponding regular subdivision. We apply these constructions to obtain a family of flat semitoric degenerations for every type A Grassmannian and complete flag variety.

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arXiv2021

Koszul algebras and Donaldson-Thomas invariants

For a given symmetric quiver $Q$, we define a supercommutative quadratic algebra $\mathcal{A}_Q$ whose Poincaré series is related to the motivic generating function of $Q$ by a simple change of variables. The Koszul duality between supercommutative algebras and Lie superalgebras assigns to the algebra $\mathcal{A}_Q$ its Koszul dual Lie superalgebra $\mathfrak{g}_Q$. We prove that the motivic Donaldson-Thomas invariants of the quiver $Q$ may be computed using the Poincaré series of a certain Lie subalgebra of $\mathfrak{g}_Q$ that can be described, using an action of the first Weyl algebra on $\mathfrak{g}_Q$, as the kernel of the operator $\partial_t$. This gives a new proof of positivity for motivic Donaldson--Thomas invariants. In addition, we prove that the algebra $\mathcal{A}_Q$ is numerically Koszul for every symmetric quiver $Q$ and conjecture that it is in fact Koszul; we also prove this conjecture for quivers of a certain class.

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arXiv2021

Totally nonnegative Grassmannians, Grassmann necklaces and quiver Grassmannians

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 cellular decomposition, whose cells are naturally labeled by Grassmann necklaces. We show that the posets of cells coincide with the reversed cell posets of the cellular decomposition of the totally nonnegative Grassmannians. We investigate algebro-geometric and combinatorial properties of these quiver Grassmannians. In particular, we describe the irreducible components, study the action of the automorphism groups of the underlying representations and describe the moment graphs. We also construct a resolution of singularities for each irreducible component; the resolutions are defined as quiver Grassmannians for an extended cyclic quiver.

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arXiv2020

Semi-infinite Plücker relations and Weyl modules

The goal of this paper is twofold. 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, i.e. the quotient by the ideal generated by the semi-infinite Plücker relations. We establish the isomorphism with the algebra of dual global Weyl modules and derive a new character formula.

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arXiv2020

Weighted PBW degenerations and tropical flag varieties

We study algebraic, combinatorial and geometric aspects of weighted 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, for example, the classical flag variety, its abelian PBW degeneration, some of its linear degenerations and a particular toric degeneration.

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arXiv2019

Peter-Weyl, Howe and Schur-Weyl theorems for current groups

The classical Peter-Weyl theorem describes the structure of the space of functions on a semi-simple algebraic group. On the level of characters (in type A) this boils down to the Cauchy identity for the products of Schur polynomials. We formulate and prove the analogue of the Peter-Weyl theorem for the current groups. In particular, in type A the corresponding characters identity is governed by the Cauchy identity for the products of q-Whittaker functions. We also formulate and prove a version of the Schur-Weyl theorem for current groups. The link between the Peter-Weyl and Schur-Weyl theorems is provided by the (current version of) Howe duality.

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arXiv2019

Large tensor products and Littlewood-Richardson coefficients

The Littlewood-Richardson coefficients describe the decomposition of tensor products of irreducible representations of a simple Lie algebra into irreducibles. Assuming the number of factors is large, one gets a measure on the space of weights. This limiting measure was extensively studied by many authors. In particular, Kerov computed the corresponding density in a special case in type A and Kuperberg gave a formula for the general case. The goal of this paper is to give a short, self-contained and pure Lie theoretic proof of the formula for the density of the limiting measure. Our approach is based on the link between the limiting measure induced by the Littlewood-Richardson coefficients and the measure defined by the weight multiplicities of the tensor products.

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arXiv2019

Linear degenerations of flag varieties: partial flags, defining equations, and group actions

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. We prove that there exists the deepest flat degeneration and the deepest flat irreducible degeneration: the former is the partial analogue of the mf-degenerate flag variety and the latter coincides with the partial PBW-degenerate flag variety. We compute the generating function of the number of orbits in the flat irreducible locus and study the natural family of line bundles on the degenerations from the flat irreducible locus. We also describe explicitly the reduced scheme structure on these degenerations and conjecture that similar results hold for the whole flat locus. Finally, we prove an analogue of the Borel-Weil theorem for the flat irreducible locus.

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arXiv2018

Symmetric Dellac configurations and symplectic/orthogonal flag varieties

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 construction is based on the quiver Grassmannian realization of the degenerate flag varieties and odd symplectic and odd and even orthogonal groups. We study basic properties of our varieties; in particular, we construct cellular decomposition in all the three cases above (as well as in the case of even symplectic group). We also compute the Poincaré polynomials in terms of certain statistics on the set of symmetric Dellac configurations.

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arXiv2018

Symmetric Dellac configurations

We define symmetric Dellac configurations as the Dellac configurations that are symmetric with respect to their centers. The symmetric Dellac configurations whose lengths are even were previously introduced by Fang and Fourier under the name of symplectic Dellac configurations, to parametrize the torus fixed points of symplectic degenerate flag varieties. In general, symmetric Dellac configurations generate the Poincaré polynomials of (odd or even) symplectic or orthogonal versions of the degenerate flag varieties. In this paper, we give several combinatorial interpretations of the polynomial extensions $(D_n(x))_{n \geq~0}$ of median Euler numbers, defined by Randrianarivony and Zeng, in terms of objects that we name extended Dellac configurations and which generate symmetric Dellac configurations. As a consequence, the cardinalities of the odd and even symmetric Dellac configurations are respectively given by the two adjoining sequences $(l_n)_{n \geq~0} = (1, 1, 3, 21, 267,\dots)$ and $(r_n)_{n \geq~0} = (1,2,10,98,1594,\dots)$, defined as specializations of the polynomials $(D_n(x))_{n \geq~0}$.

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arXiv2018

Vertex algebras and coordinate rings of semi-infinite flags

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 associated graded space with respect to the dual filtration is isomorphic to the homogenous coordinate ring of semi-infinite flag variety. We describe the ring structure in terms of vertex operators and endow the homogenous coordinate ring with a structure of $P/Q$-graded vertex operator algebra. We use the vertex algebra approach to derive semi-infinite Plücker-type relations in the homogeneous coordinate ring.

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arXiv2017

Representation theoretic realization of non-symmetric Macdonald polynomials at infinity

We study the nonsymmetric Macdonald polynomials specialized at infinity from various points of view. First, we define a family of modules of the Iwahori algebra whose characters are equal to the nonsymmetric Macdonald polynomials specialized at infinity. Second, we show that these modules are isomorphic to the dual spaces of sections of certain sheaves on the semi-infinite Schubert varieties. Third, we prove that the global versions of these modules are homologically dual to the level one affine Demazure modules.

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arXiv2017

Generalized Weyl modules, alcove paths and Macdonald polynomials

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 quantum Bruhat graph. We make use of the Orr-Shimozono formula in order to prove that the $t=\infty$ specializations of the nonsymmetric Macdonald polynomials are equal to the characters of certain generalized Weyl modules.

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arXiv2017

Vertices of FFLV polytopes

FFLV polytopes describe monomial bases in irreducible representations of $\mathfrak{sl}_n$ and $\mathfrak{sp}_{2n}$. We study various sets of vertices of FFLV polytopes. First, we consider the special linear case. We prove the locality of the set of vertices with respect to the type $A$ Dynkin diagram. Then we describe all the permutation vertices and after that we describe all the simple vertices and prove that their number is equal to the large Schröder number. Finally, we derive analogous results for symplectic algebras.

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