Nakajima Quiver Varieties Research Articles

Stable envelopes for slices of the affine Grassmannian

The affine Grassmannian associated to a reductive group G\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$${\ extbf{G}}$$\\end{document} is an affine analogue of the usual flag varieties. It is a rich source of Poisson varieties and their symplectic resolutions. These spaces are examples of conical symplectic resolutions dual to the Nakajima quiver varieties. We study the cohomological stable envelopes of Maulik and Okounkov (Astérisque 408:ix+209, 2019) in this family. We construct an explicit recursive relation for the stable envelopes in the G=PSL2\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$${\ extbf{G}}= \ extbf{PSL}_{2}$$\\end{document} case and compute the first-order correction in the general case. This allows us to write an exact formula for multiplication by a divisor.

Crepant Resolutions of Stratified Varieties via Gluing

Abstract Let $X$ be a variety with a stratification ${\mathcal{S}}$ into smooth locally closed subvarieties such that $X$ is locally a product along each stratum (e.g., a symplectic singularity). We prove that assigning to each open subset $U \subset X$ the set of isomorphism classes of locally projective crepant resolutions of $U$ defines an ${\mathcal{S}}$-constructible sheaf of sets. Thus, for each stratum $S$ and basepoint $s \in S$, the fundamental group acts on the set of germs of projective crepant resolutions at $s$, leaving invariant the germs extending to the entire stratum. Global locally projective crepant resolutions correspond to compatible such choices for all strata. For example, if the local projective crepant resolutions are unique, they automatically glue uniquely. We give criteria for a locally projective crepant resolution $\rho : \tilde X \to X$ to be globally projective. We show that the sheafification of the presheaf $U \mapsto \operatorname{Pic}(\rho ^{-1}(U)/U)$ of relative Picard classes is also constructible. The resolution is globally projective only if there exist local relatively ample bundles whose classes glue to a global section of this sheaf. The obstruction to lifting this section to a global ample line bundle is encoded by a gerbe on the singularity $X$. We show the gerbes are automatically trivial if $X$ is a symplectic quotient singularity. Our main results hold in the more general setting of partial crepant resolutions, that need not have smooth source. We apply the theory to symmetric powers and Hilbert schemes of surfaces with du Val singularities, finite quotients of tori, multiplicative, and Nakajima quiver varieties, as well as to canonical three-fold singularities.

On the vertex functions of type A quiver varieties

The goal of this paper is to better understand the quasimap vertex functions of type A Nakajima quiver varieties. To that end, we construct an explicit embedding of any type A quiver variety into a type A quiver variety with all framings at the rightmost vertex of the quiver. Then, we consider quasimap counts, showing that the map induced by this embedding on equivariant K-theory preserves vertex functions.

Harish-Chandra bimodules for type A rational Cherednik algebras

We study Harish-Chandra bimodules for the rational Cherednik algebra associated to the symmetric group Sn. In particular, we show that for any parameter c∈C, the category of Harish-Chandra Hc-bimodules admits a fully faithful embedding into the category Oc, and describe the irreducibles in the image. We also construct a duality on the category of Harish-Chandra bimodules, and in fact we do this in a greater generality of quantizations of Nakajima quiver varieties. We use this duality, along with induction and restriction functors, to describe, as an abelian category, the smallest Serre subcategory of the category of Harish-Chandra bimodules containing the regular bimodule, as well as to explicitly describe the tensor products of its irreducible objects.

Namikawa-Weyl groups of affinizations of smooth Nakajima quiver varieties

We give a description of the Namikawa-Weyl group of affinizations of smooth Nakajima quiver varieties based on combinatorial data of the underlying quiver, and compute some explicit examples. This extends a result of McGerty and Nevins for quiver varieties associated to Dynkin quivers.

Nakajima's quiver varieties and triangular bases of rank-2 cluster algebras

Berenstein and Zelevinsky introduced quantum cluster algebras [3] and the triangular bases [4]. The support conjecture in [12] asserts that the support of a triangular basis element for a rank-2 cluster algebra is bounded by an explicitly described region that is possibly concave. In this paper, we prove the support conjecture for all skew-symmetric rank-2 cluster algebras.

On categories 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 A ¯ λ ( n , ℓ ) \overline {\mathcal {A}}_{\lambda }(n, \ell ) if both dim ⁡ V = n \operatorname {dim}V=n and the number of loops ℓ \ell are greater than 1 1 . We show that when n ≤ 3 n\leq 3 there is a Hamiltonian torus action with finitely many fixed points, provide the dimensions of Hom-spaces between standard objects in category O \mathcal {O} and compute the multiplicities of simples in standards for n = 2 n=2 in case of one-dimensional framing and generic one-parameter subgroups. We establish the abelian localization theorem and find the values of parameters, for which the quantizations have infinite homological dimension.

Singular quiver varieties over extended dynkin quivers

Abstract 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 parameters and use this to construct a number of hyper-Kähler cobordisms between binary polyhedral spaces.

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.

Elliptic Stable Envelopes of Affine Type A Quiver Varieties

Abstract We generalize Smirnov’s formula for the elliptic stable envelopes of the Hilbert scheme of points in $\mathbb {C}^2$ to the case of affine type $A$ Nakajima quiver varieties constructed with positive stability condition. We allow for arbitrary choices of polarization and a fairly general choice of chamber. This paper is a companion to the Maple code developed by the author, which implements the formulas described in this paper.

Quantum difference equation for Nakajima varieties

For an arbitrary Nakajima quiver variety X, we construct an analog of the quantum dynamical Weyl group acting in its equivariant K-theory. The correct generalization of the Weyl group here is the fundamental groupoid of a certain periodic locally finite hyperplane arrangement in $${{\,\mathrm{Pic}\,}}(X)\otimes {\mathbb {C}}$$ . We identify the lattice part of this groupoid with the operators of quantum difference equation for X. The cases of quivers of finite and affine type are illustrated by explicit examples.

SHUFFLE ALGEBRAS FOR QUIVERS AND R-MATRICES

AbstractWe define slope subalgebras in the shuffle algebra associated to a (doubled) quiver, thus yielding a factorization of the universal R-matrix of the double of the shuffle algebra in question. We conjecture that this factorization matches the one defined by [1, 18, 32, 33, 34] using Nakajima quiver varieties.

Capped vertex with descendants for zero dimensional A∞ quiver varieties

In this paper, we study the capped vertex functions associated to certain zero-dimensional type-A Nakajima quiver varieties. The insertion of descendants into the vertex functions can be expressed by the Macdonald operators, which leads to explicit combinatorial formulas for the capped vertex functions.We determine the monodromy of the vertex functions and show that it coincides with the elliptic R-matrix of symplectic dual variety. We apply our results to give the vertex functions and the characters of the tautological bundles on the quiver varieties formed from arbitrary stability conditions.

Hamiltonian reduction for affine Grassmannian slices and truncated shifted Yangians

Generalized affine Grassmannian slices provide geometric realizations for weight spaces of representations of semisimple Lie algebras. They are also Coulomb branches, symplectic dual to Nakajima quiver varieties. In this paper, we prove that neighboring generalized affine Grassmannian slices are related by Hamiltonian reduction by the action of the additive group. We also prove a weaker version of the same result for their quantizations, algebras known as truncated shifted Yangians.

Nakajima quiver varieties, affine crystals and combinatorics of Auslander-Reiten quivers

We obtain an explicit crystal isomorphism between two realizations of crystal bases of finite dimensional irreducible representations of simple Lie algebras of type A and D. The first realization we consider is a geometric construction in terms of irreducible components of certain Nakajima quiver varieties established by Saito and the second is a realization in terms of isomorphism classes of quiver representations obtained by Reineke. We give a homological description of the irreducible components of Lusztig's quiver varieties which correspond to the crystal of a finite dimensional representation and describe the promotion operator in type A to obtain a geometric realization of Kirillov-Reshetikhin crystals.

Euler continuants in noncommutative quasi-Poisson geometry

AbstractIt was established by Boalch that Euler continuants arise as Lie group valued moment maps for a class of wild character varieties described as moduli spaces of points on$\mathbb {P}^1$by Sibuya. Furthermore, Boalch noticed that these varieties are multiplicative analogues of certain Nakajima quiver varieties originally introduced by Calabi, which are attached to the quiver$\Gamma _n$on two vertices andnequioriented arrows. In this article, we go a step further by unveiling that the Sibuya varieties can be understood using noncommutative quasi-Poisson geometry modelled on the quiver$\Gamma _n$. We prove that the Poisson structure carried by these varieties is induced, via the Kontsevich–Rosenberg principle, by an explicit Hamiltonian double quasi-Poisson algebra defined at the level of the quiver$\Gamma _n$such that its noncommutative multiplicative moment map is given in terms of Euler continuants. This result generalises the Hamiltonian double quasi-Poisson algebra associated with the quiver$\Gamma _1$by Van den Bergh. Moreover, using the method of fusion, we prove that the Hamiltonian double quasi-Poisson algebra attached to$\Gamma _n$admits a factorisation in terms ofncopies of the algebra attached to$\Gamma _1$.

Derived representation schemes and Nakajima quiver varieties

We introduce a derived representation scheme associated with a quiver, which may be thought of as a derived version of a Nakajima variety. We exhibit an explicit model for the derived representation scheme as a Koszul complex and by doing so we show that it has vanishing higher homology if and only if the moment map defining the corresponding Nakajima variety is flat. In this case we prove a comparison theorem relating isotypical components of the representation scheme to equivariant K-theoretic classes of tautological bundles on the Nakajima variety. As a corollary of this result we obtain some integral formulas present in the mathematical and physical literature since a few years, such as the formula for Nekrasov partition function for the moduli space of framed instantons on S^4. On the technical side we extend the theory of relative derived representation schemes by introducing derived partial character schemes associated with reductive subgroups of the general linear group and constructing an equivariant version of the derived representation functor for algebras with a rational action of an algebraic torus.

Quot Schemes for Kleinian Orbifolds

For a finite subgroup $\Gamma\subset {\mathrm{SL}}(2,\mathbb{C})$, we identify fine moduli spaces of certain cornered quiver algebras, defined in earlier work, with orbifold Quot schemes for the Kleinian orbifold $[\mathbb{C}^2/\Gamma]$. We also describe the reduced schemes underlying these Quot schemes as Nakajima quiver varieties for the framed McKay quiver of $\Gamma$, taken at specific non-generic stability parameters. These schemes are therefore irreducible, normal and admit symplectic resolutions. Our results generalise our previous work on the Hilbert scheme of points on $\mathbb{C}^2/\Gamma$; we present arguments that completely bypass the ADE classification.

Punctual Hilbert schemes for Kleinian singularities as quiver varieties

For a finite subgroup $\Gamma\subset \mathrm{SL}(2,\mathbb{C})$ and $n\geq 1$, we construct the (reduced scheme underlying the) Hilbert scheme of $n$ points on the Kleinian singularity $\mathbb{C}^2/\Gamma$ as a Nakajima quiver variety for the framed McKay quiver of $\Gamma$, taken at a specific non-generic stability parameter. We deduce that this Hilbert scheme is irreducible (a result previously due to Zheng), normal, and admits a unique symplectic resolution. More generally, we introduce a class of algebras obtained from the preprojective algebra of the framed McKay quiver by a process called cornering, and we show that fine moduli spaces of cyclic modules over these new algebras are isomorphic to quiver varieties for the framed McKay quiver and certain non-generic choices of stability parameter.

A study of irreducible components of Springer fibers using quiver varieties

It is a remarkable theorem by Maffei–Nakajima that the Slodowy variety, which is a subvariety of the resolution of the nilpotent cone, can be realized as a Nakajima quiver variety of type A. However, the isomorphism is rather implicit as it takes to solve a system of equations in which the variables are linear maps. In this paper, we construct solutions to this system under certain assumptions. This establishes an explicit and efficient way to compute the image of a complete flag contained in the Slodowy variety under the Maffei–Nakajima isomorphism and describe these flags in terms of quiver representations. As Slodowy varieties contain Springer fibers naturally, we can use these results to provide an explicit description of the irreducible components of two-row Springer fibers in terms of a family of kernel relations via quiver representations.