Questions tagged [quiver-varieties]
Quiver variety refers to a number of varieties constructed as moduli spaces of representations of quivers.
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Riemann-Hilbert problem via quiver description
The moduli space of Fuchsian systems over $\mathbb{P}^1$ with prescribed adjoint orbits conditions at poles a.k.a. additive Deligne-Simpson problem can be presented under purely quiver description.The ...
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Do local and global symplectic resolutions have same monodromy?
Yoshinori Namikawa associates a Weyl group $ W $ to any symplectic affine complex variety $ X $ with good $ \mathbb{C}^* $-action. He provides a semi-explicit description of $ W $, which requires ...
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geometric objects in quiver variety corresponding to short exact sequences
I was studying quiver variety and known that representations of a quiver correspond to points in the corresponding quiver variety. So if give you a fixed triple representations $(M_1,M_2,M_3)$, I was ...
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Chainsaw quiver variety and parabolic bundle
How can we relate chainsaw quiver varieties with ADE type Nakajima quiver varieties?
We know that we can obtain ADE type quiver varieties (instantons over ALE spaces) by taking $\Gamma$ equivariant ...
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additive vs multiplicative quiver/hypertoric varieties - properties
It is a standard fact that a smooth Nakajima quiver variety / hypertoric variety $X$ has the following properties:
It is holomorphic symplectic $(X,\omega)$, in fact 1'. hyperkahler
It has a ...
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Hyperkähler structure of framed instantons over $\smash{\overline{\mathbb{C}P}}^2$
When underlying $4$ manifold is compact and hyperkähler, the philosophy of infinite dimensional moment map tells us that its instantons moduli space is also hyperkähler. I'm curious about the ...
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Two notions of stability
Let $Q$ be a finite quiver (i.e. an oriented graph). A representation of $Q$ is by definition a module over the path algebra of $Q$. More concretely, a representation associates to every vertex $v \in ...
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Getting an equivariant morphism
Let $X\subset\mathbb{A}^n$ be an irreducible affine variety over an algebraically closed field $\mathbb{K}$ of characteristic zero. Suppose we have two linearly reductive algebraic groups $G$, $G'$ ...
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How to determine if an invariant rational function is defined at the $\theta$-polystable point
Background:
Let $A$ be a finite-dimensional (associative and unital) algebra over $\mathbb{C}$. Assume there is a quiver $Q=(Q_0,Q_1)$, where $Q_0$ is the set of vertices and $Q_1$ is the set of ...
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Example of an irreducible component with an open set of infinitely many codimension 2 (codimension 3) orbits
Let $\mathbb{K}$ be an algebraically closed field of characteristics $0$. Let $A$ be a finite dimensional (associative and unital) algebra over $\mathbb{K}$. Assume there is a quiver $Q=(Q_0,Q_1)$, ...
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Is $U\subseteq X^{s}(L)$?
Let $X$ be an irreducible affine variety over an algebraically closed field $\mathbb{K}$ of characteristics $0$. Let $G$ be a connected, linearly reductive, affine algebraic group acting regularly on $...
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Non-empty stable locus of an irreducible component
I have a vague question:
Let $X$ be an algebraic pre-scheme and $G$ be a linear reductive group. Consider the G.I.T. quotient $X{/\!/}G$. Is there any result (maybe in some special case) which tells ...
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Drinfel'd polynomials for evaluation representations of $\mathbf{U}_q(\mathbf{L}\mathfrak{g})$?
We know for type $A$, there is an evaluation homomorphism from quantum affine algebra to quantum algebra,
$$\operatorname{ev}_a:\mathbf{U}_q(\mathbf{L}\mathfrak{g})\to \mathbf{U}_q(\mathfrak{g})$$
for ...
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Nakajima quiver varieties for ADE quiver with one dimensional framing
Let $Q$ be a quiver of type $ADE$, $I$ is the set of vertices of $Q$. Let $\mathfrak{M}({\mathbf{v}},{\mathbf{w}})$ be a Nakajima quiver variety for such quiver (here ${\mathbf{v}}=(v_i)_{i \in I}$ is ...
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Absolutely indecomposable objects and moduli space
In the setting of moduli spaces of vector bundles/quiver representations etc I've encountered a few times situations like the one that I'll try to explain thereafter .I was wondering if there's a &...
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Moduli stack of quiver representations
Let $Q$ be a finite quiver. As far as I know there's a great amount of work concerning the so-called quiver varieties one can associate to it. Loosely speaking, these are obtained by taking GIT ...
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(Super)integrable systems on quiver varieties
In recent papers
https://arxiv.org/abs/2101.05520
https://arxiv.org/abs/2001.06911
(super)integrable systems on quiver varieties for cyclic and comet-shaped quivers are constructed.
My question: are ...
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Nakajima reflection functors and the flavour/framing group action
Nakajima has constructed so-called reflection functors that are isomorphisms between different quiver varieties that have the same framing $\mathbf{w}:$
$$\Phi_{\sigma}:\mathfrak{M}_\zeta(Q,\mathbf{v}...
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Intuition for the McGerty-Nevins compactification of quiver varieties
In Section 4 of the paper Kirwan surjectivity for quiver varieties (Inventiones Math. 2018) McGerty and Nevins define a compactification of the moduli space of representations
of the preprojective ...
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Which abelian categories have homological dimension 1?
In this MSRI lecture Geometry of Quiver Varieties I, Victor Ginzburg describes all abelian categories of homological dimension $1$ as being either
a category of representations $\mathrm{Rep}_\mathbf{...
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Image of a quiver variety under natural morphism
We know that the natural morphism $\pi:\mathfrak{M}_{\theta}(Q,\mathbf{v},\mathbf{w})\rightarrow \mathfrak{M}_0(Q,\mathbf{v},\mathbf{w})$ between a smooth and affine quiver variety is not necessarily ...
3
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Core of the Jordan quiver variety
It is known that, given the Jordan quiver, dimension vectors
$\textbf{v}=n,\textbf{w}=1$ and a stability condition $\theta<0,$ the corresponding quiver variety $\mathcal{M}_{\theta}(n,1)\cong \...
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Core components of quiver varieties as fiber bundles of flag varieties
Is there an example of Nakajima quiver variety of type A which has all core components smooth, such that at least one of them is NOT an iterated fibre bundle of flag manifolds (i.e. a space obtained ...
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Lagrangian cores of quiver variety in different GIT chambers
Let $\pi: \mathfrak{M}_{(\theta,0)}(Q,\text{v},\text{w}) \rightarrow \mathfrak{M}_{(0,0)}(Q,\text{v},\text{w})$ be the projective morphism between Nakajima quiver varieties when complex moment ...
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Quiver variety analogue of Grothendieck-Springer resolution
A standard example of Nakajima quiver varieties are type A Springer resolutions $\widetilde{\mathcal{N}} \to \mathcal{N}$. In the theory of Springer resolutions it is often beneficial to consider the ...
3
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Generalizations of toric varieties
I would like to know, whether the following quotient construction has been considered, or whether it makes sense:
One way to think of toric varieties is as a quotient of $\mathbb{C}^n$ (minus ...
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How is the free modular lattice on 3 generators related to 8-dimensional space?
Here are three facts which sound potentially related. What are the actual relationships?
In 1900, Dedekind constructed the free modular lattice on 3 generators as a sublattice of the lattice of ...
6
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For which quiver varieties is Kirwan surjectivity known?
The cohomology of Nakajima quiver varieties is a quite interesting object. It's equipped with some natural classes given by the Chern classes of the tautological bundles associated to the spaces in ...
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Conditions under which a bijective morphism of quasi-projective varieties is an isomorphism
I'm currently reading a paper by Nakajima (Quiver Varieties and Tensor Products), and I'm having a hard time understanding a very specific step in his proof of Lemma 3.2. Essentially, we have two (...
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Flag Varieties via Quiver Varieties
In type $A$ it is possible to realise the flag variety $\mathcal{F}$ of $\text{SL}_{n}(\mathbb{C})$ via Nakajima's quiver varieties: consider the vectors $v=(1,2,\ldots, n-1), w=(0,\ldots,0,n)$. Then, ...
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Non-symmetric quiver varieties
Given a symmetric Cartan datum $(I,\cdot)$, H. Nakajima has defined a family of varieties - known as quiver varieties - and has used them to give geometric constructions of the representation theory ...