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Questions tagged [quiver]

A quiver is an oriented graph which might contain multiple edges and loops. The terminology is used in representation-theory of finite dimensional algebras, where one considers functors from this graph, viewed as a category, to the category of vector spaces.

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Definition of directed graph in Robertson & Seymour's article n°23

I am interested in theorem 1.7 in Robertson and Seymour's article Graph minors XXIII. It states : Let $\Omega$ be a well-quasi-order. For $i\leq 1$ let $G_i$ be a directed graph, and let $\phi_i : V (...
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Computing Auslander Reiten Quivers using Computer Algebra software

I was wondering if there is any algebra software that can be used to compute the Auslander Reiten graph for a given path algebra, and find out related details like the exterior algebra structure etc.
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Local finite dimensional algebras are $\tau$-tilting finite

Let $A$ be a finite dimensional algebra over an algebraically closed field $\mathbb{K}$. If $A$ is a local algebra (has unique maximal left ideal), then it is well known that $A$ is $\tau$-tilting ...
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Weyl group of $D_n$

We have that the root system of the Dynkin diagram of type $D_n$ can be realized by considering the standard inner product in $\mathbb{R}^n$ and the vectors $\{e_1 - e_2, \ldots, e_{n-1} - e_n, e_{n-1}...
Andreadel1988's user avatar
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Does the dual of a quiver representation correspond to the dual of the associated $\mathbb{K}Q$-module?

So, this question comes from the theory of quiver representations (more concretely, from the book "An introduction to Quiver Representations" by Harm Derksen and Jerzy Weyman). Let me lay ...
Duarte Costa's user avatar
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How to Understand the Decompositions of Representations of $A_n$.

In short, I would like a thorough walk-through as to how representations of $A_{n}$ are decomposed from the pure linear algebra perspective. For example, a representation of an $A_2$ quiver is a ...
jdogz1912's user avatar
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Matrix of the minimal projective presentation of a $\tau$-rigid module

Let $A$ be a finite dimensional algebra over an algebraically closed field $\mathbb{K}$ of characteristics zero. Suppose $A$ is given by the bound quiver $(Q,I)$. We will use $P_l$ to denote the ...
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Use quiver representation to study integral extension of Artinian rings

I have some ideas about the integral extension of Artinian rings denoted by $R \subset S$. Since $\mathrm{Spec}(R)$ is finite and discrete, I construct a category $\mathscr{R}$ whose objects are ...
Siyuan Yin's user avatar
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Dimers from Postnikov diagrams

I am reading several papers on Postnikov diagrams, dimer models, and quivers. From the Postnikov diagram, we can draw a dimer model and an ice quiver. My question is as follows. Does every dimer model(...
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Factoring out the socle of the projective-injectives for a quiver algebra

Let $A$ be a quiver given in the GAP-package QPA (https://folk.ntnu.no/oyvinso/QPA/) . Question: Is there a fast/easy way to obtain $A/soc(U)$ (using QPA), where $U$ is the direct sum of all ...
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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 ...
It'sMe's user avatar
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indecomposable representation of a quiver example

Let be https://i.imgur.com/hufWpv4.jpg a quiver. We have that $\mathbb{K}Qe_2=span \{e_2,a'',a''' \}$, so $V_1=e_1\mathbb{K}Qe_2=\{0 \}$, $V_2=e_2\mathbb{K}Qe_2=span \{e_2 \}$, $V_3=a''\mathbb{K}Qe_2=...
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What is quiver of Quivers?

I made a triangular-shaped quiver q (with three vertices and three arrows) and gave representations to the quiver, that is these representations are objects of $Vect^{Free(q)}$. now I was trying to ...
Rabia Sagheer's user avatar
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Diagrams as visualizations of functors

Can this quiver $A\to B\to C$ be a diagram in some category? (Let's hope that now my question is "specific" so the community bot won't close it) This is the background for me asking this: my ...
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Quivers as the underlying structure of categories

The structure of a small category is a quiver - every vertex of the quiver represents an object with its identity morphism, and the edges of the quiver represent the non trivial morphisms (I know you ...
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Need help understanding proof of Proposition 2.9 from Quiver representations by Schiffler.

Prop. 2.9 says the following: A representation $Q$ is simple iff it is isomorphic to $S(i)$ for some $i \in Q_0$. The proof given in the text starts off as follows: It is clear that the $S(i)$ are ...
John Smith's user avatar
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Endomorphism ring of a quiver representation over an algebraically closed field.

I'm in trouble with an exercise. The text asks to consider a quiver with only one loop, a field $k = \mathbb{F}_2(t)$, and a representation $V$ given by the matrix: \begin{bmatrix} 0 & t \\ 1 &...
Marco Bagnara's user avatar
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How do Calabi-Yau algebras arise from cyclic quivers?

In his famous preprint Calabi-Yau algebras arXiv:math/0612139v3 [math.AG], Ginzburg seems to state that the Jacobian algebra of a cyclic quiver is Calabi-Yau algebra of dimension 3. There is a quote ...
Noto_Ootori's user avatar
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The existence of an indecomposable summand of a projective module such that it is isomorphic to a given simple module

I am reading the book “ Elements of the Representation Theory of Associative Algebras:Volume 1” written by Ibrahim Assem, Daniel Simson, Andrzej Skowronski. I have a question about the proof of ...
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A question about Nakayama functor in Elements of the Representation Theory of Associative Algebras:Volume 1

I am reading the book “ Elements of the Representation Theory of Associative Algebras:Volume 1” written by Ibrahim Assem, Daniel Simson, Andrzej Skowronski and I have a question about chapter 2 ...
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Isotropic vectors of Tits form

Let $\Gamma=(\Gamma_0,\Gamma_1)$ be a multigraph, that is, $\Gamma_0$ is a (finite) set of vertices and $\Gamma_1$ is a (finite) multiset of edges. This means we allow multiple edges between two ...
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The underlying vector space of the completed path algebra

I'm looking at the following question, whose answer I disagree with: How to understand completed path algebra? For context, let $Q = (V, A)$ be a quiver with infinitely many arrows. Then it is often ...
Jakub Waniek's user avatar
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Standard projective resolution of module of the path algebra

I have the quiver $Q$ with two vertices $v_1,v_2$ and two arrows $a_1,a_2$ that go from $v_1$ to $v_2$. Let $kQ$ be the path algebra of my quiver, where $k$ is a field, and let $M=kQ/\langle a_1\...
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Minimally sufficient condition for composability of quiver morphisms?

Given three quivers $(X_0, X_1, \sigma, \tau)$, $(Y_0, Y_1, \phi, \psi)$, and $(Z_0, Z_1, \chi, \omega)$, and two morphisms $F := (F_0 : X_0 \to Y_0, F_1 : X_1 \to Y_1) : (X_0, X_1, \sigma, \tau) \to (...
Jos van Nieuwman's user avatar
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Dimension formula/algorithm for quiver varieties?

The title says the question. For a quiver and a dimension vector and a stability vector, we can construct a moduli space of semistable quiver representations with the given dimension vector. The ...
Display Name's user avatar
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Jordan and Dynkin quivers

I have a simple question. If I am not wrong, we can define $\tilde{A}_l$ affine Dynkin quiver of type A, for $l\geq1$. It has $l+1$ vertices that we can order from $0$ to $l$ such that $i$ is ...
wood's user avatar
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Adjoint functors induce in representations of quivers? (references)

Let $Q$ be a quiver and ${C}_1, C_2$ be two (small) categories. Let $F:C_1\rightarrow C_2$ and $G:C_2\rightarrow C_1$ be two (covariant) functors. Is it true that if $F$ and $G$ induce functors $$ ...
IMP's user avatar
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Endomorphism ring of a quiver representation

Consider the quiver with one loop, and consider the representation on $\mathbb{F}_{2}(t)$ described by the matrix $$W=\begin{pmatrix} 1 & t \\ 1 & 0 \end{pmatrix}.$$ I need to calculate the ...
Martin Gale's user avatar
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Correspondence between type A Nakajima Quiver varieties and weight spaces in the tensor product of fundamental representations of $\mathfrak{gl}(N)$

In the paper "New Quiver-Like varieties and Lie Superalgebras" by Rimanyi and Rozansky it is claimed that "There is a well-known correspondence between an $A_n$-type framed Nakajima ...
staedtlerr's user avatar
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Showing $J(A)=0$, where $A$ is the path algebra of an oriented cycle

Let $Q$ be a cyclic quiver with $n$ vertices and consider the path algebra $A=KQ$. I want to show that the Jacobson radical of $A$ is zero. Since the Jacobson radical is the intersection of all ...
Sir Socket's user avatar
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Path algebra on quivers of type ADE

Let $K$ be a field. For $n\in \mathbb{N}$ consider the equioriented $A_n$-quiver. It is easy to see that the path algebra $K[A_n]$ is isomorphic as a $K$-algebra to the algebra of upper triangular $n\...
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Connected components of $Q(\mathrm{s\tau\textrm{-}tilt} A)$

$\newcommand{\hy}{\textrm{-}}$ I'm reading about support $\tau$-tilting modules and their mutations. I'm trying to understand the mutation quiver. Let $A$ be a finite dimensional algebra over an ...
It'sMe's user avatar
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Fixed points of subtorus in Nakajima quiver variety (Lemma 3.2 in Nakajima's Quiver varieties and tensor products)

I am having trouble understanding the proof of lemma 3.2 in Nakajima's paper "Quiver varieties and tensor products," available on arxiv. This is where he shows that the fixed components of a ...
staedtlerr's user avatar
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$\text{soc}(V)$ and $\text{rad}(V)$ of a representation $V$ of an acyclic quiver $Q$

Let $Q$ be an acyclic quiver, and $V = (V_i, V_a)$ a representation of $Q$. I want to show that $$\text{soc}(V) = \bigoplus_{i \in Q_0} \left(\bigcap_{a \in Q_1, s(a) = i} \ker(V_a) \right), \hspace{...
LinearAlgebruh's user avatar
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Some reference requests for path algebras

I can find some properties and characterizations of path algebras and its elements in representation theory books but I want some references that study and treat path algebras as invertible elements, ...
The Student's user avatar
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1 answer
95 views

Classify the finite-dimensional indecomposable representations of the quiver $A_n$

I have to classify the finite-dimensional indecomposable representations of the quiver $A_n$ up to isomorphism: 1 $\longrightarrow$ 2 $\longrightarrow \cdots \longrightarrow$ n Any idea of how to ...
Mikel's user avatar
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How does the center of a Path Algebra KQ for a quiver Q look like?

) I am attending a Foundation Course in Representation Theory and I am struggling with the following problem: determine the center of the path algebra KQ for a quiver Q. First of all I've thought that ...
Mikel's user avatar
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Applications of representation theory in topology

I'm beginning to study representation theory, in particular, quiver representations. Since I have more familiarity with topology, I was wondering if there is any applications of these things to ...
Lucas Henrique's user avatar
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1 answer
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How to decompose representations of $1\to 2\leftarrow 3$?

For now let's consider the $n=2$-subspace quiver: $Q = \bullet \to \bullet \leftarrow \bullet$. I have shown (with the help of Kirillov) that to classify these representations, it suffices to classify ...
staedtlerr's user avatar
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Intuition for representation of a quiver inherited from the representation of the quiver's path algebra

Let $V$ be a representation of the path algebra $P_{Q}$. From this representation, we can construct a representation of $Q$ as follows: let $V_i =p_i V$, and for any edge $h$, let $x_h = a_h|_{p_h'} v ...
Cathartic Encephalopathy's user avatar
2 votes
2 answers
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Dimension of $(U+V+W)$

Let $U, V, W$ be subspaces of a vector space. I know that in general, the equation $\dim(U+V+W)=\dim U+\dim V+\dim W-\dim(U \cap V)-\dim (U\cap W)-\dim (V \cap W)+\dim (U \cap V \cap W)$ doesn't hold. ...
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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 ...
It'sMe's user avatar
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Why is a linear representation of a quiver a functor?

Let $Q:=(Q_0,Q_1,s,t)$ be a quiver and $k$ some field. A representation $(M,\rho)$ of $Q$ over $k$ is the following data: A $k$-vector space $M_v$ for every $v\in Q_0$; A $k$-linear map $\rho(a):M_{...
user831160's user avatar
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References for Coefficient Quivers

I would like to study about Coefficient Quivers, but I cannot find a good reference. I could find many papers working with coefficient quivers, but none of them give a book or a "initial" ...
IMP's user avatar
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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 $Q=(Q_0,Q_1)$, ...
It'sMe's user avatar
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Use the equivalence between representations and $kQ$-modules to find $\mathrm{rad}(P(i))$ and $\mathrm{top}(P(i))$

Question: I had the exercise below (not graded) and I think I have all the components to solve it, but somehow I can't seem to find a proper proof. To what right $kQ$-modules do I have to relate $P(i)$...
anonymous's user avatar
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Example of a brick-infinite, tame, triangular algebra of global dimension$\geq3$

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 ...
It'sMe's user avatar
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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\...
It'sMe's user avatar
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McKay correspondence for Irreps of $G \subset SU(2)$

We follow Alexander Kirilov's book "Quiver Representations and Quiver Varieties", Section 8.3: McKay correspondence: Let $G$ be a nontrivial finite subgroup in $SU(2)$. Let $Q(G)$ be the ...
user267839's user avatar
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2 votes
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Notation for Modules Over Quiver Algebras

Disclaimer: I know that my question has probably answers in textbooks, but I have only found the answer for quiver representations, and not for modules over quiver algebras. I know that these two ...
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