Questions tagged [quiver-varieties]

Quiver variety refers to a number of varieties constructed as moduli spaces of representations of quivers.

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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 ...
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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 ...
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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 ...
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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 ...
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