Quivers of type A, flag varieties and representation theory

Etingof conjecture for quantized quiver varieties II: affine quivers

We study the representation theory of quantizations of Gieseker moduli spaces. Namely, we prove the localization theorems for these algebras, describe their finite dimensional representations and two-sided ideals as well as their categories O in some special cases. We apply this to prove our conjecture with Bezrukavnikov on the number of finite dimensional irreducible representations of quantized quiver varieties for quivers of affine type.

光滑 Nakajima 箭簇簇仿射化的 Namikawa-Weyl 群

Namikawa-Weyl groups of affinizations of smooth Nakajima quiver varieties

本文利用底层箭簇的组合数据刻画光滑 Nakajima 箭簇簇仿射化的 Namikawa-Weyl 群，并通过显式计算将 McGerty–Nevins 的 Dynkin 情形结果推广到更一般情形。

Modules with 1-dimensional socle and components of Lusztig quiver varieties in type A

We study modules with 1-dimensional socle for preprojective algebras for type A quivers. In particular, we classify such modules, determine all homomorphisms between them, and then explain how they may be used to describe the components of Lusztig quiver varieties.

A class of perverse sheaves on framed representation varieties of the Jordan quiver

A class of perverse sheaves on framed representation varieties of the Jordan quiver is defined. Its relationship with product of symmetric groups, tensor product of Schur algebras, and tensor product of Fock spaces are addressed.

Quiver varieties and the character ring of general linear groups over finite fields

We describe the "generic" part of the character ring of general linear groups over a finite field in terms of quiver representations.

Dynkin箭图的箭图Grassmannian的旗版本没有奇上同调

Flag versions of quiver Grassmannians for Dynkin quivers have no odd cohomology

本文证明了Dynkin箭图（A、D、E型）的箭图Grassmannian旗版本无奇上同调，A和D型具仿射铺砌，并推广了刚性表示的对角分解结果。

Degenerate flag varieties of type A and C are Schubert varieties

We show that in type A or C any degenerate flag variety is in fact isomorphic to a Schubert variety in an appropriate partial flag manifold.

Quiver Grassmannians and Auslander varieties for wild algebras

Let k be an algebraically closed field and A a finite-dimensional k-algebra. Given an A-module M, the set G_e(M) of all submodules of M with dimension vector e is called a quiver Grassmannian. If D,Y are A-modules, then we consider Hom(D,Y) as a B-module, where B is the opposite of the endomorphism ring of D, and the Auslander varieties for A are the quiver Grassmannians of the form G_e Hom(D,Y). Quiver Grassmannians, thus also Auslander varieties are projective varieties and it is known that every projective variety occurs in this way. There is a tendency to relate this fact to the wildness o…

Quiver Varieties and Path Realizations arising from Adjoint Crystals of type $A_n^{(1)}$

Let $B(Λ_0)$ be the level 1 highest weight crystal of the quantum affine algebra $U_q(A_n^{(1)})$. We construct an explicit crystal isomorphism between the geometric realization $\mathbb{B}(Λ_0)$ of $B(Λ_0)$ via quiver varieties and the path realization ${\mathcal P}^{\rm ad}(Λ_0)$ of $B(Λ_0)$ arising from the adjoint crystal $\adjoint$.

Quiver varieties and Hall algebras

In this paper we give a geometric construction of the quantum group Ut(G) using Nakajima categories, which were developed in [29]. Our methods allow us to establish a direct connection between the algebraic realization of the quantum group as Hall algebra by Bridgeland [1] and its geometric counterpart by Qin [24].

Etingof conjecture for quantized quiver varieties

We compute the number of finite dimensional irreducible modules for the algebras quantizing Nakajima quiver varieties. We get a lower bound for all quivers and vectors of framing and provide an exact count in the case when the quiver is of finite type or is of affine type and the framing is the coordinate vector at the extending vertex. The latter case precisely covers Etingof's conjecture on the number of finite dimensional irreducible representations for Symplectic reflection algebras associated to wreath-product groups. We use several different techniques, the two principal ones are categor…

Quiver flag varieties and multigraded linear series

This paper introduces a class of smooth projective varieties that generalise and share many properties with partial flag varieties of type A. The quiver flag variety M_\vartheta(Q,r) of a finite acyclic quiver Q (with a unique source) and a dimension vector r is a fine moduli space of stable representations of Q. Quiver flag varieties are Mori Dream Spaces, they are obtained via a tower of Grassmann bundles, and their bounded derived category of coherent sheaves is generated by a tilting bundle. We define the multigraded linear series of a weakly exceptional sequence of locally free sheaves E …

$G$-stable pieces and partial flag varieties

We will use the combinatorics of the $G$-stable pieces to describe the closure relation of the partition of partial flag varieties in \cite[section 4]{L3}.

具有极小支撑的量子化花箭簇簇上的Harish-Chandra双模

Harish-Chandra bimodules over quantized flower quiver varieties with minimal support

本研究对具有极小支撑的量子化花箭簇簇上的Harish-Chandra双模进行分类，证明维数向量为$n$时，仅当整数量化参数足够大或足够小时存在$n!$个极小支撑单Harish-Chandra双模，核心工具为增强版限制函子。

GKM-Theory for Torus Actions on Cyclic Quiver Grassmannians

We define and investigate algebraic torus actions on quiver Grassmannians for nilpotent representations of the equioriented cycle. Examples of such varieties are type $\tt A$ flag varieties, their linear degenerations, finite dimensional approximations of the $GL_n$-affine flag variety and affine Grassmannian. We show that these quiver Grassmannians, equipped with our torus action, are GKM-varieties and that their moment graph admits a combinatorial description in terms of coefficients quiver of the underlying quiver representations. By adapting to our setting results by Gonzales, we are able …

Translation quiver varieties

We introduce a framework of translation quiver varieties which includes Nakajima quiver varieties as well as their graded and cyclic versions. An important feature of translation quiver varieties is that the sets of their fixed points under toric actions can be again realized as translation quiver varieties. This allows one to simplify quiver varieties in several steps. We prove that translation quiver varieties are smooth, pure and have Tate motivic classes. We also describe an algorithm to compute those motivic classes.

A型箭袋簇、环路格拉斯曼流形与幂零锥的比较

Comparison of quiver varieties, loop Grassmannians and nilpotent cones in type A

本文在A型情形下建立Nakajima带标架箭袋簇、矩阵共轭类与环路格拉斯曼流形之间的显式几何等价，并由此实现紧化、分解及$(GL(m),GL(n))$对偶的几何化。

箭图簇与秩3中的根重数

Quiver varieties and root multiplicities in rank 3

本文利用箭图簇为秩3对称Kac-Moody代数的虚根空间维数提供组合上界，并给出二部Dynkin图下的显式提取方法及计算证据。

On categories $\mathcal{O}$ of quiver varieties overlying the bouquet graphs

We study representation theory of quantizations of Nakajima quiver varieties associated to bouquet quivers. We show that there are no finite dimensional representations of the quantizations $\overline{\mathcal{A}}_λ(n, \ell)$ if dim $V=n$ is greater than $1$ and so is the number of loops $\ell$. We find that there is a Hamiltonian torus action with finitely many fixed points in case $n\leq 3$, provide the dimensions of Hom-spaces between standard objects in category $\mathcal{O}$ and compute the multiplicities of simples in standards for $n=2$ in case of one-dimensional framing and generic one…

Modular perverse sheaves on flag varieties III: positivity conditions

We further develop the general theory of the "mixed modular derived category" introduced by the authors in a previous paper in this series. We then use it to study positivity and Q-Koszulity phenomena on flag varieties.

Equivariant Sheaves on Flag Varieties

We show that the Borel-equivariant derived category of sheaves on the flag variety of a complex reductive group is equivalent to the perfect derived category of dg modules over the extension algebra of the direct sum of the simple equivariant perverse sheaves. This proves a conjecture of Soergel and Lunts in the case of flag varieties.

Singular Quiver Varieties over Extended Dynkin Quivers

We classify the singularities in the unframed Nakajima quiver varieties associated with extended Dynkin quivers and the corresponding minimal imaginary root with a small restriction on the parameter and use this to construct a number of hyper-Kähler cobordisms between binary polyhedral spaces.

Desingularizations of quiver Grassmannians via graded quiver varieties

Inspired by recent work of Cerulli-Feigin-Reineke on desingularizations of quiver Grassmannians of representations of Dynkin quivers, we obtain desingularizations in considerably more general situations and in particular for Grassmannians of modules over iterated tilted algebras of Dynkin type. Our desingularization map is constructed from Nakajima's desingularization map for graded quiver varieties.

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, …

Fermionic forms and quiver varieties

We prove a formula relating the fermionic forms and the Poincare polynomials of quiver varieties associated to a finite quiver. Applied to quivers of type ADE, our result implies a version of the fermionic conjecture of Lusztig.

Generalized quiver varieties and triangulated categories

In this paper, we introduce generalized quiver varieties which include as special cases classical and cyclic quiver varieties. The geometry of generalized quiver varieties is governed by a finitely generated algebra P: the algebra P is self-injective if the quiver Q is of Dynkin type, and coincides with the preprojective algebra in the case of classical quiver varieties. We show that in the Dynkin case the strata of generalized quiver varieties are in bijection with the isomorphism classes of objects in proj P, and that their degeneration order coincides with the Jensen-Su-Zimmermann's degener…

Quiver Grassmannians and degenerate flag varieties

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 an injective representation of a Dynkin quiver. It is proven that these are (typically singular) irreducible normal local complete intersection varieties, which admit a group action with finitely many orbits, and a cell…

Type D quiver representation varieties, double Grassmannians, and symmetric varieties

We unify aspects of the equivariant geometry of type $D$ quiver representation varieties, double Grassmannians, and symmetric varieties $GL(a+b)/GL(a)\times GL(b)$; in particular we translate results about singularities of orbit closures, combinatorics of orbit closure containment, and torus equivariant $K$-theory between these three families. These results are all obtained from our generalization of a construction of Zelevinsky for type $A$ quivers to the type $D$ setting. More precisely, we give explicit embeddings with nice properties of homogeneous fiber bundles over type $D$ quiver repres…

Quiver varieties and Beilinson-Drinfeld Grassmannians of type A

We construct Nakajima's quiver varieties of type A in terms of conjugacy classes of matrices and (non-Slodowy's) transverse slices naturally arising from affine Grassmannians. In full generality quiver varieties are embedded into Beilinson-Drinfeld Grassmannians of type A. Our construction provides a compactification of Nakajima's quiver varieties and a decomposition of an affine Grassmannian into a disjoint union of quiver varieties. As an application we provide a geometric version of skew and symmetric $(GL(m), GL(n))$ duality.

Coherent Sheaves on Quiver Varieties and Categorification

We construct geometric categorical Lie algebra actions on the derived category of coherent sheaves on Nakajima quiver varieties. These actions categorify Nakajima's construction of Kac-Moody algebra representations on the K-theory of quiver varieties. We define an induced affine braid group action on these derived categories.

On quiver varieties and affine Grassmannians of type A

We construct Nakajima's quiver varieties of type A in terms of affine Grassmannians of type A. This gives a compactification of quiver varieties and a decomposition of affine Grassmannians into a disjoint union of quiver varieties. Consequently, singularities of quiver varieties, nilpotent orbits and affine Grassmannians are the same in type A. The construction also provides a geometric framework for skew $(GL(m), GL(n))$ duality and identifies the natural basis of weight spaces in Nakajima's construction with the natural basis of multiplicity spaces in tensor products which arises from affine…

Smooth quiver quotient varieties

In this paper we classify all the quivers and corresponding dimension vectors having a smooth space of semisimple representation classes. The result is that these quiver settings can be reduced via some specific reduction steps to 3 simple types.

Comet-shaped quiver varieties, Weyl group actions, and modified Kostka polynomials

We study an algebra spanned by modified Kostka polynomials. Particular structure coefficients of this algebra are interpreted as traces of some Weyl group actions on the intersection cohomology of comet-shaped quiver varieties.

Lectures on Nakajima's Quiver Varieties

This is an expanded version of lectures given at a Summer School "Geometric methods in Representation Theory" (Grenoble, 2008).

Derived Blow-ups and Birational Geometry of Nested Quiver Varieties

Given a quiver, Nakajima introduced the quiver variety and the Hecke correspondence, which is a closed subvariety of Cartesian products of quiver varieties. In this paper, we consider two nested quiver varieties as fiber products of Hecke correspondences along natural projections. After blowing up the diagonal, we prove that they are isomorphic to a quadruple moduli space which Neguţ observed for the Jordan quiver. The main ingredient is Jiang's derived projectivization theory and Hekking's derived blow-up theory, while we obtain the local model when both the two derived schemes are quasi-smoo…

Linear degenerations of flag varieties

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, generalizing the corresponding results on degenerate flag varieties. To study normality, cell decompositions of quiver Grassmannians are constructed in a wider context of equioriented quivers of type A.

On the Irreducibility of Some Quiver Varieties

We prove that certain quiver varieties are irreducible and therefore are isomorphic to Hilbert schemes of points of the total spaces of the bundles $\mathcal O_{\mathbb P^1}(-n)$ for $n \ge 1$.

具有重数的幂零箭图簇与可对称化晶体

Nilpotent quiver varieties with multiplicities and symmetrisable crystals

该研究构造带重数箭图的幂零箭图簇类似物，通过开发新稳定性条件及利用与赫克对应关系的作用，将斋藤对卡茨 - 穆迪代数晶体的几何构造从对称扩展到可对称化情形。

Quiver varieties and Lusztig's algebra

We study preprojective algebras of graphs and their relationship to module categories over representations of quantum SL(2). As an application, ADE quiver varieties of Nakajima are shown to be subvarieties of the variety of representations of a certain associative algebra introduced by Lusztig.

光滑的斜线对应于淬火品种

Smooth correspondences between quiver varieties

本文提出了一种新的光滑对应于淬火品种，用于构建奇点解析并证明其不可约性和Cohen-Macaulay性质，扩展了之前关于希尔伯特方案的研究。

Arithmetic harmonic analysis on character and quiver varieties II

We study connections between the topology of generic character varieties of fundamental groups of punctured Riemann surfaces, Macdonald polynomials, quiver representations, Hilbert schemes on surfaces, modular forms and multiplicities in tensor products of irreducible characters of finite general linear groups.

Homological approach to the Hernandez-Leclerc construction and quiver varieties

In a previous paper the authors have attached to each Dynkin quiver an associative algebra. The definition is categorical and the algebra is used to construct desingularizations of arbitrary quiver Grassmannians. In the present paper we prove that this algebra is isomorphic to an algebra constructed by Hernandez-Leclerc defined combinatorially and used to describe certain graded Nakajima quiver varieties. This approach is used to get an explicit realization of the orbit closures of representations of Dynkin quivers as affine quotients.

四维Nakajima标量簇

Nakajima quiver varieties in dimension four

本文通过组合方法对四维Nakajima标量簇进行了分类，并反驳了关于G₄商奇异性能否作为quiver簇的疑问。

拟射影簇是箭图表示的完全正合子范畴的格拉斯曼流形

Quasi-projective varieties are Grassmannians for fully exact subcategories of quiver representations

本文通过限制Reineke同构到完全正合子范畴，证明拟射影簇可表示为箭图格拉斯曼流形。

Quiver varieties and Frenkel-Kac construction

An affine Lie algebra acts on cohomology groups of quiver varieties of affine type. A Heisenberg algebra acts on cohomology groups of Hilbert schemes of points on a minimal resolution of a Kleinian singularity. We show that in the case of type $A$ the former is obtained by Frenkel-Kac construction from the latter.

Kac's conjecture from Nakajima quiver varieties

We prove a generating function formula for the Betti numbers of Nakajima quiver varieties. We prove that it is a q-deformation of the Weyl-Kac character formula. In particular this implies that the constant term of the polynomial counting the number of absolutely indecomposable representations of a quiver equals the multiplicity of a a certain weight in the corresponding Kac-Moody algebra, which was conjectured by Kac in 1982.

A categorical action on quantized quiver varieties

In this paper, we describe a categorical action of any Kac-Moody algebra on a category of quantized coherent sheaves on Nakajima quiver varieties. By "quantized coherent sheaves," we mean a category of sheaves of modules over a deformation quantization of the natural symplectic structure on quiver varieties. This action is a direct categorification of the geometric construction of universal enveloping algebras by Nakajima.

On the equations and classification of toric quiver varieties

Toric quiver varieties (moduli spaces of quiver representations) are studied. Given a quiver and a weight there is an associated quasiprojective toric variety together with a canonical embedding into projective space. It is shown that for a quiver with no oriented cycles the homogeneous ideal of this embedded projective variety is generated by elements of degree at most $3$. In each fixed dimension $d$ up to isomorphism there are only finitely many $d$-dimensional toric quiver varieties. A procedure for their classification is outlined.

Vector bundles on projective varieties and representations of quivers

We present equivalences between certain categories of vector bundles on projective varieties, namely cokernel bundles, Steiner bundles, syzygy bundles, and monads, and full subcategories of representations of certain quivers. As an application, we provide decomposability criteria for such bundles.

Quiver varieties and adjoint crystals of level $\ell$ for type $A_n^{(1)}$

Let $B(Λ)$ be a level $\ell$ highest weight crystal of the quantum affine algebra $U_q(A_n^{(1)})$. We construct an explicit crystal isomorphism between the geometric realization $\gB(Λ)$ of the crystal $B(Λ)$ using quiver varieties and the path realization $\mathcal{P}^{\rm ad}(Λ)$ arising from the adjoint crystal $\Bad{\ell}$ of level $\ell$.

格雷德准旗变分与迪金克准旗导出范畴中的退化

Degenerations in graded quiver varieties and in derived categories of Dynkin quivers

本文研究了迪金克准旗导出范畴中退化现象，揭示了格雷德准旗变分与导出范畴退化之间的对应关系。

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.

Graded quiver varieties and singularities of normalized R-matrices for fundamental modules

We present a simple unified formula expressing the denominators of the normalized R-matrices between the fundamental modules over the quantum loop algebras of type ADE. It has an interpretation in terms of representations of the Dynkin quivers and can be proved in a unified way using the geometry of graded quiver varieties. As a by-product, we obtain a geometric interpretation of Kang-Kashiwara-Kim's generalized quantum affine Schur-Weyl duality functor when it arises from a family of fundamental modules. We also study several cases when the graded quiver varieties are isomorphic to the graded…

Schubert Quiver Grassmannians

Quiver Grassmannians are projective varieties parametrizing subrepresentations of given dimension in a quiver representation. We define a class of quiver Grassmannians generalizing those which realize degenerate flag varieties. We show that each irreducible component of the quiver Grassmannians in question is isomorphic to a Schubert variety. We give an explicit description of the set of irreducible components, identify all the Schubert varieties arising, and compute the Poincaré polynomials of these quiver Grassmannians.