All Questions
Tagged with quivers ag.algebraic-geometry
46 questions
6
votes
1
answer
139
views
Quiver variety, generically symplectic
Theorem 11.3.1 (iv) of "Noncommutative Geometry and Quiver algebras" by Crawley-Boevey, Etingof and Ginzburg claims that, for a dimension vector $\mathbf{d}\in\Sigma_0$, the quiver variety $...
- 985
3
votes
0
answers
102
views
Projective resolution of a quiver with relations
How do we compute the projective resolution of a representation of a quiver with relations.
For example consider the Beilinson quiver $B_4$
$.
with the relations $\{\alpha_j^k\alpha_i^{k-1}=\alpha_i^...
- 179
1
vote
0
answers
130
views
A basis of the weight space in the semi-invariant ring corresponding to the weight $\langle(2,3,2),\cdot\rangle$
I'm trying to understand Example 10.11.1 on page 225 of the book "An introduction to quiver representations" by Harm Derksen and Jerzy Weyman (see the attached screenshot below)
I want to ...
- 839
0
votes
0
answers
71
views
"Approximating" ring of semi-invariants
I'm trying to calculate the semi-invariant ring for certain types of quivers. For a very brief introduction to semi-invariant rings of quiver please have a look at this wikipedia article at the ...
- 839
1
vote
0
answers
257
views
Confusion regarding the invariant rational functions
I was reading S. Mukai's- An Introduction to Invariants and Moduli, and I came across Proposition 6.16 on page 195 (see the screenshot below)
It says that "every invariant rational function can ...
- 839
1
vote
0
answers
80
views
When is $Y$ not an orbit closure?
Let $G$ be a linearly reductive algebraic group acting regularly on an affine space over $\mathbb{A}^n$ an algebraically closed field $\mathbb{K}$. Let $Y$ be a $G$-invariant (closed) affine ...
- 839
2
votes
0
answers
143
views
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 ...
- 373
1
vote
0
answers
208
views
Is $\langle\chi,\lambda\rangle=0$, whenever the limit exists? Where is the mistake?
Suppose $G$ is a linearly reductive algebraic group acting linearly on a finite dimensional vector space $V$ over $\mathbb{C}$. This induces an action on the coordinate ring $\mathbb{C}[V]$ (see here)....
- 839
2
votes
0
answers
129
views
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 $G$, $G'$ ...
- 839
1
vote
0
answers
52
views
Is the Schofield semi invariant defined at $V/IV$?
Let $A=\mathbb{K}Q$ be the path algebra of an acyclic quiver $Q$ over an algebraically closed field $\mathbb{K}$, and $0\not=I\subset\mathbb{K}Q$ be an admissible ideal. Let $W$ be a left $A$-module ...
- 839
2
votes
1
answer
186
views
Orbits in the open set given by Rosenlicht's result
Let $G$ be a linearly reductive algebraic group, and let $X$ be an irreducible affine variety, over an algebraically closed field $\mathbb{K}$, with a regular action of $G$. By Rosenlicht's result, we ...
- 839
1
vote
0
answers
127
views
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 ...
- 839
4
votes
0
answers
259
views
Road map for learning cluster algebras
I'm a PhD student and I would like learn about cluster algebras. I'm wondering what is a good reference (i.e., has detailed explanations, examples, etc) to learn from the basic and what are some of ...
- 839
3
votes
0
answers
111
views
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 $Q=(Q_0,Q_1)$, ...
- 839
1
vote
0
answers
95
views
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 $...
- 839
1
vote
0
answers
128
views
Example of a brick-infinite, tame, triangular algebra of global dimension$\geq 3$
I'm trying to compute some examples and I'm unable to come up with a following example:
What is(are) the example(s) of an acyclic quiver $Q$ with relations such that the 2-Kronecker quiver is NOT a ...
- 839
1
vote
0
answers
115
views
Prove that $B$ is a directing module
Let $A\cong\mathbb{K}Q/I$ be a finite dimensional, associative, basic $\mathbb{K}$ algebra, where $\mathbb{K}$ is algebraically closed and $Q$ is a finite Gabriel quiver on $n$ vertices and $I\...
- 839
2
votes
0
answers
268
views
Understanding a proof of a result of Schofield
I'm reading a paper of Aidan Schofield- "General Representations of Quivers" and I'm trying to understand the proof of Theorem 3.3. I'm having trouble understanding the argument that's ...
- 839
6
votes
1
answer
312
views
Prove that $\overline{a}_{11}$ is a prime element in $R$
Consider the affine space given by four $2\times 2$ matrices, i.e., $\mathbb{A}^{16}\cong M(\mathbb{C})_{2\times 2}^4$. Now, consider the algebraic set $V$ given by the vanishing of the relation $AB-...
- 839
1
vote
0
answers
144
views
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 ...
- 839
1
vote
1
answer
305
views
Quiver varieties associated to D_4
Let $Q=(I,\Omega)$ be the $D_4$ affine quiver. We choose as dimension vector $(2,1,1,1,1)$ (where $2$ is on the central vertex). As this dimension vector is indivisible, we can choose a generic $\...
- 1,061
14
votes
2
answers
1k
views
Invariants of matrices (by simultaneous $\mathrm{GL}_n$ conjugation) over arbitrary rings
$\DeclareMathOperator\GL{GL}$Let $R$ be a commutative ring, let $R[n] := R[M_d^{\oplus n}]$ be the polynomial ring on $nd^2$ variables corresponding to the coordinates of $n$-many $d\times d$ matrices....
- 7,527
4
votes
2
answers
869
views
Research topics in representation theory of algebras [closed]
I was wondering what are some of the hot topics in quiver representation or representation theory of algebras that can lead to good mathematics and is important to many mathematicians and top ...
5
votes
0
answers
351
views
What representation theoretic properties does the semi-invariant ring tell us?
I'm asking this question as a continuation of discussion and answer given by Hugh Thomas at the following post: Why do people study semi-invariant ring (in general)?
I have been studying about semi-...
- 839
12
votes
0
answers
347
views
Quivers as noncommutative curves
I've heard that an idea behind noncommutative geometry (in dim 1) is to study "noncommutative" analogues of $\text{Coh}(\text{curve})$, rather than the curve directly. Apparently the ...
- 5,701
12
votes
1
answer
577
views
Embedding of a derived category into another derived category
I am considering the following two cases:
Assume that there is an embedding: $D^b(\mathcal{A})\xrightarrow{\Phi} D^b(\mathbb{P}^2)$and the homological dimension of $\mathcal{A}$ is equal to $1$($\...
- 1,982
5
votes
0
answers
273
views
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 ...
- 2,964
3
votes
1
answer
177
views
$M^{ss}_{(2,2)}(K_3,(-1,1))$ is isomorphic to $M_{\mathbb{P}^2}(0,2)$
Suppose $K_3$ is the Kronecker quiver with 3 arrows, and $M^{ss}_{(2,2)}(K_3,(-1,1))$ is the moduli space of semi stable representation of dimension $(2,2)$ wrt the weight $(-1,1)$. It is claim in the ...
- 815
21
votes
5
answers
3k
views
Why are coherent sheaves on $\Bbb P^1$ derived equivalent to representations of the Kronecker quiver?
I'm looking for an explanation or a reference to why there is this equivelence of triangulated categories: $${D}^b(\mathrm {Coh}(\Bbb P^1))\simeq {D}^b(\mathrm {Rep}(\bullet\rightrightarrows \bullet))$...
- 1,185
22
votes
2
answers
2k
views
Quiver representations and coherent sheaves
I've heard that under certain assumptions on an algebraic variety $X$ there exist a quiver $Q$ for which there is an equivalence $$D^b(\mathsf{Coh}(X))\simeq D^b(\mathsf{Rep}(Q))$$ between the ...
user95283
4
votes
1
answer
192
views
Quiver invariants as polynomials/algebraic curves
I'm interested in algebraic curves one can associate to gauge or string theories. Examples involve Seiberg-Witten curves or family of A-polynomials which define holomorphic Lagrangian submanifolds for ...
- 243
5
votes
0
answers
146
views
Injectivity of a standard map in quiver representation
Let $X$ be a smooth projective variety, and assume its divisor class group is finite and free. Let $E_1,E_2,\ldots,E_n$ be line bundles on $X$. Define $L_k=E_1+\ldots E_k$, and let $Q$ be the ...
- 815
9
votes
1
answer
1k
views
Reference request: tangent space to moduli space of coherent sheaves is $\operatorname{Ext}^1(E, E)$
Is there a standard reference for the fact that, in an appropriate algebraic-geometrical context, the tangent space at the point $[E]$ to the moduli space $\mathcal M$ is something like $\operatorname{...
- 1,980
5
votes
1
answer
509
views
analog of Lusztig nilpotent scheme
Fix a quiver $Q$ without loop. Denote the set of vectices of $Q$ by $I$.
Let $\Lambda_V$ be the Lusztig nilpotent scheme with associated vector space $V$ over $I$. Briefly speaking, when $Q$ is a $ADE$...
- 849
2
votes
1
answer
501
views
Second cohomology groups of Nakajima quiver varieties
Let $X=M(v,w)$ be a Nakajima quiver variety for a quiver $Q$. Can one calculate the second singular cohomology groups $H^2(X,\mathbb Z)$ or $H^2(X,\mathbb C)$ explicitly, and if not, are there some ...
- 2,964
18
votes
1
answer
566
views
Is there a cotangent bundle of a stable $\infty$-category?
Let $C$ be a stable $\infty$-category. Is there any categorical construction $C \mapsto T^* C$, where $T^* C$ is another stable $\infty$-category, that specializes to the following?
When $C$ is the ...
- 4,949
7
votes
1
answer
341
views
How can one show that orbit closures in representations of a linear quiver don't have small resolutions?
Let $1\to \cdots\to n$ be a linear quiver of length $n$. Let $\mathbf{d}=(d_1,\dots,d_n)$ be a dimension vector. It's well known (for example, by Gabriel's theorem, but also by basic linear algebra) ...
- 44.7k
6
votes
2
answers
917
views
Do fixed point sets in equivariant crepant resolutions have the same cohomology? How about for the specific case of Nakajima quiver varieties?
A crepant resolution $f:Y\to X$ is a resolution of singularities with $f^*(K_X)=K_Y$. Crepant resolutions do not always exist, and when they exist they may not be unique. However, different crepant ...
- 2,372
8
votes
1
answer
1k
views
Quiver varieties and the affine Grassmannian
Recently I was watching a talk: http://media.cit.utexas.edu/math-grasp/Ben_Webster.html and at the end the lecturer gave a correspondence (I was having trouble with subscripts so changed the notation ...
- 741
3
votes
0
answers
238
views
Invariant Subvarieties of Variety of Quiver Representations
I'd like to understand a special case of the following rather general algebraic geometry question:
Given an algebraic group $G$ acting on a variety $V$, can we describe the $G$-invariant subvarieties ...
- 31
23
votes
4
answers
3k
views
Deformations of Nakajima quiver varieties
Are deformations of Nakajima quiver varieties also Nakajima quiver varieties ?
In case the answer to this is (don't k)no(w), here are some simpler things to ask for.
(If you're a differential ...
- 1,191
5
votes
0
answers
638
views
Hirzebruch-Riemann-Roch for quiver varieties?
These days, I attended a workshop at North Carolina State University. The key lecturer is Professor Nakajima. He introduced two types of quiver variety. One of them is affine, another one is quasi-...
- 5,515
12
votes
3
answers
1k
views
construct scheme from quivers?
I heard from some guys working in noncommutative geometry talking about the idea that one can construct the noncommutative space from quivers. I feel it is rather interesting. However, I can not image ...
- 1,305
11
votes
1
answer
792
views
What's known about the stalks of Lusztig's perverse sheaves on quiver varieties?
Lusztig has defined a category of perverse sheaves on the moduli space of representations of a Dynkin quiver (see his paper) corresponding to canonical basis vectors.
I'm interested in the stalks ...
- 44.7k
11
votes
2
answers
2k
views
What do the local systems in Lusztig's perverse sheaves on quiver varieties look like?
In "Quivers, perverse sheaves and quantized enveloping algebras," Lusztig defines a category of perverse sheaves on the moduli stack of representations of a quiver. These perverse sheaves are defined ...
- 44.7k
4
votes
4
answers
1k
views
Near Trivial Quiver Varieties
So, today I started learning the definition of a quiver variety, and wanted to make sure I'm understanding things right, so first, my setup:
I've been looking at the simplest case that didn't look ...
- 16k