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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The category of finite dimensional right $KG$-modules is equivalent to the category of finite dimensional representations of a quiver $Q$

Let $K$ be an algebraically closed field and $G$ be a finite group such that $|G|$ is not divisible by the characteristic of $K$ (so that Maschke's theorem can be applied). Let $Q$ be the quiver ...
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Systematically labeling arrows in GAP's QPA

I'm trying to build a quiver in QPA which has 42 nodes, labeled by some of the signvectors in $\{\pm 1\}^6$. I want to build a rule that assigns an arrow $a\to b$ if $a$ and $b$ differ in exactly one ...
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39 views

Why is a morphism of quiver representations a subset of this product of morphisms?

Given two representations $M,N$ of a quiver Q, why is the set of all morphisms of representations from $M$ to $N$ a subspace of $\prod_{i\in Q_0} \mathrm{Hom}(M,N)$? I mean why the product of $\mathrm{...
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Extension-algebras of $A_4$

Consider the quiver $Q\colon 1\xrightarrow{\alpha} 2\xrightarrow{\beta} 3\xrightarrow{\gamma} 4$ and the algebra $A=k[Q]/(\gamma\beta\alpha)$. Denote the simple $A$-modules by $L(-)$ and let $M$ be ...
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Are these the projective and injective representations of these quivers?

Find all indecomposable projective and indecomposable injective representations of these two quivers over the field $k$ up to isomorphism. I've drawn my answer in this picture. Can you please check ...
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How do we factorize the path algebra by the relation?

I read Representation Theory of Artin Algebras by Auslander, Reiten and Smalo and I have a question about this example. Can you please explain why does the basis of $\Lambda \bar{e_1}$ contain $\bar{\...
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48 views

Showing that two path algebra quotients aren't isomorphic

I've been working through Auslander, Reiten, and Smalø's Representation Theory of Artin Algebras, and I've gotten stuck on Exercise III.8(c). The larger exercise has me consider the following quiver $...
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57 views

Topological Invariance in Data Structures

I need to do a PhD in Pure Mathematics and I am thinking of Topological Data Analysis. I want to use persistent homology and quiver representation to obtain topological features in data structures. ...
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Indecomposble Direct Summand of Nilpotent Endomorphism of Quiver Representation

I was reading the proof of Gabriel's Theorem (Theorem 8.12) from Ralf Schiffler's 'Quiver Representations'. The proof begins by showing that for an indecomposable module $M$, the vector space $Hom(M,M)...
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On the Variety of Representations of a Fixed Dimension Vector

I have been trying to understand the proof of Gabriel's Theorem from Ralf Schiffler's 'Quiver Representations'. In chapter 8, he has defined, for a finite quiver $Q=(Q_0,Q_1)$ without oriented cycles ...
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Topological Group Structure on Endomorphism Group of a Quiver Representation and Relations with Dimensions of Varieties

I have been reading chapter 8 of Ralf Schiffler's 'Quiver Representations'. There he has defined, for a finite quiver $Q=(Q_0,Q_1)$ without oriented cycles and for a (fixed) dimension vector $\textbf{...
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Variety of Quiver Representations

I have been reading chapter 8 of Ralf Schiffler's 'Quiver Representations', where he proves Gabriel's Theorem characterizing connected quivers with finitely many isoclasses of indecomposable ...
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Generating relations of an admissible ideal are not uniquely determined

Let $Q=(Q_0,Q_1)$ be a quiver as shown below, where $Q_0=\{1,2\}$, and $Q_1=\{\alpha,\beta,\gamma\}$, $\alpha:1\to 1$, $\beta:1\to 2$, $\gamma:2\to 1$. I would like to show that two admissible ideals $...
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Bound quiver representation of matrix algebras

I have been trying to write a bound quiver representation of each of the following algebras: $$A=\begin{bmatrix} K&0&0&0&0\\ K&K&0&0&0\\ K&0&K&0&0\\ K&...
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Every indecomposable representation is a brick if $Q$ has no cycles

I'm currently following a course on quiver representations, and stumbled across the definition of a brick, which is a quiver representation $M$ with the property that $\mathrm{End}(M)\cong k$, where $...
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Maximal ideals in the path algebra of a quiver

Let $Q = (Q_0,Q_1)$ be a finite quiver and let $kQ$ be its path algebra, that is the algebra of formal linear combinations of paths on $Q$ with multiplication given by concatenation of paths. I think ...
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Decomposition of quiver representation with Jordan cell map

I am reading Persistence Theory: From Quiver Representations to Data Analysis by Steve Y. Oudot and have the following question on Gabriel's theorem. Consider the quiver $$\bullet \longrightarrow \...
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$KQ$ acylic connected finite dimension path algebra. $Cok(\oplus_{x\in Q_0}Hom_k(M_x,M'_x)\to\oplus_{a\in Q_1}Hom_k(M_{s(a)},M'_{t(a)}))=Ext^1(M,M')$

Let $Q=(Q_0,Q_1,s,t)$ be an acylic connected finite quiver where $Q_0$ are vertices, $Q_1$ are arrows, and $s,t:Q_1\to Q_0$ are source and target respectively. Let $K$ be a field. Let $KQ$ be the ...
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Looking for an example of finite acyclic quiver path algebra with a simple module of infinite projective dimension

I am looking for an example of finite acyclic quiver path algebra with a simple module of infinite projective dimension. I have considered the following. Consider quiver $1\to 2\to 3\to 4$. Form path ...
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An admissible numbering is always an admissible sequence of sinks-proof.

Let $Q$ be a finite, connected, and acyclic quiver. We have that: there exists a bijection between $Q_0$ (set of vertices of $Q$) and the set $\{1,\dots,n\}$ such that if we have an arrow $i \to j$,...
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An admissible numbering of an acyclic quiver

Let $Q$ be an acyclic quiver. In here, we read that there exists a bijection between $Q_0$ (set of vertices of $Q$) and the set $\{1,\dots,n\}$ such that if we have an arrow $i \to j$, then $j >...
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A Request for Clarification on Quiver Reps and Gabriel's Theorem

I've turned myself around thinking about indecomposable representations of quivers and Gabriel's Theorem and I would be grateful if someone could point out my misunderstanding. Suppose you have a $...
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Path algebra $A=kQ$ is not isomorphic to $A^{op}$

This is an exercise from An Introduction to Quiver Representations by Harm Derksen and Jerzy Weyman. Exercise 1.6.3. Let $Q$ be the quiver \begin{equation*} \circ \longrightarrow \circ \...
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Are diagrams of a quiver the same as small diagrams

I've just started to wrap my head around category theory, and came across two (from my perspective not obviously equivalent) definitions of a (small) diagram in a category $\mathcal{C}$: Definition 1 ...
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Morphisms from injective to preprojective modules for non-Dynkin quivers

Let $Q$ be a quiver without oriented cycles. Suppose in addition that $Q$ is non-Dynkin. I am looking for the reference or proof of the following fact: Let $I$ be an injective $kQ$-module and $M$ be ...
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48 views

Generating arbitrarily large indecomposable quiver representations

Consider a quiver $Q$ which is not of (euclidean or affine) ADE-type, i.e., $Q$ has wild representation type. Question: How can I generate an indecomposable $Q$-representation $M$ of arbitrarily ...
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Stability conditions for a quiver

Let $Q$ be the quiver with vertices $I = \{i_1,i_2,i_3,i_4,j\}$ and for each $k$ one arrow $i_k \to j$. Let $\Theta$ the function on $\Bbb Z I$ defined by $\Theta(i_k) = 0, \Theta(j) = 1$. ...
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Explanation of $Theta$ semi stable points (or representation)

I try to understand the precise meaning of a $\Theta$-semi-stable set of points. Say in the context of quiver representations: Let $Q = (Q_0, Q_1)$ be a quiver $(V_i, \alpha_{i,j})$ a representation ...
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Reflection Functor of a Quiver

Fix a Quiver $Q=(Q_{0},Q_{1})$, where $Q_{0}$ is the set of vertices and $Q_{1}$ is the set of edges. For $j\in Q_{0}$ define $\sigma_{j}Q$ to be the quiver in which all that include the vertex $j$ ...
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Are valued-quiver species and combinatorial species related?

In this paper, Joel Lemay defines a species as a generalization of a valued quiver $Q$, where to each node $i \in Q_0$ we assign a division ring $\mathbf{k}_i$, and to each arrow $(a \colon i \to j) \...
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What's the term to describe a morphism that isn't a composite of morphisms?

Is there a term to describe a morphism in a category that cannot be written as a composition of two (non-identity) morphisms? Naively Googling around I haven't found anyone talking about this, so I ...
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Is there a natural way to regard a valued quiver as a category?

You can think of a quiver, a collection of vertices $Q_0$ and arrows $Q_1$, as a free category with objects $Q_0$ and morphisms generated by the morphisms of $Q_1$. More precisely this is a functor ...
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What kind of Categorical object is an RDF Model?

RDF1.1 semantics defines an RDF Interpretation and mixes up the model and the mapping from the syntax to the model. I want to work out independently what its category of all models is. (In order to ...
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What is the (co)limit of a Quiver?

Given that a Quiver (also known as directed multigraphs) is so important in category theory, being that every Quiver produces a free Category, and every Category has an underlying Quiver, I was ...
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List of path arrows in GAP

I am using the QPA package with GAP and have the following problem: I want to, given a path $p$, obtain a list (preferably ordered) of the arrows which make up $p$. For example, if I have arrows $...
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Computing Hochschild cohomology of an algebra in GAP

I would like to calculate Hochschild cohomology of a path algebra of a quiver (modulo some relations) using GAP. Creating the quiver and its path algebra modulo relations is explained in detail in ...
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Choosing different term orders for Groebner basis calculations in GAP

I am using the QPA and GBNP packages in GAP to analyze path algebra quotients. I use the GBNP package for computing Groebner bases for the ideals in the path algebras, and the term orders for these ...
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37 views

admissible ideal of affine rank 3 acyclic quiver $\tilde{A}_{12}$

Does affine rank 3 acyclic quiver $\tilde{A}_{12}$ with arrows $2 \to 1$, $3\to 2$ and $3 \to 1$ has admissible ideal? For any finite quiver $Q$, a two-sided ideal $I$ of $KQ$ is said to be ...
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Problem with hom-spaces and their dimensions in GAP

I am running the following example in GAP using the QPA package: ...
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Usage and Realization of a Quiver Representation.

From my understanding, a quiver is basically a finite directed graph (a multidigraph). And a representation assigns a vector space to each vertex, and a linear map to each arrow. Formally, $Q = (Q_0, ...
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Can this puzzle be solved using the representation theory of quivers?

This riddle originates in the youtube video here. It's mathematical content was summarised here as follows: There's a $5\times 5$ grid of nodes, all nodes are (bidirectionally) connected to their ...
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Full projective resolutions for path algebras in GAP

I am trying to do various computations on path algebras using GAP. One such computation is obtaining projective resolutions of simples modules (and possibly other modules). The command in the GAP ...
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98 views

Indecomposable representation of an acyclic quiver on 3 vertices

I have a question: Suppose we have the quiver \begin{matrix} & 1 & \overset{}{\longrightarrow} && 2 \\ && \searrow_{} & & \downarrow_{} \\ & & &&...
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From a lower triangular matrix to its quiver representation

I have a question about quivers: Suppose we have an arbitrary lower triangular matrix algebra $A= \{\begin{pmatrix} a & 0 & 0\\ c & b & 0\\ e&d&a \end{pmatrix}: a , b, c, d \...
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Mistake in the proof of Theorem 2.24 of Quiver Representations by Ralf Schiffler?

I’m currently reading through Ralf Schiffler’s Quiver Representations, but I’m having trouble with the proof of the theorem below. In the following, $Q$ denotes a finite, acyclic quiver with set of ...
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Significance of adjoint relationship with Ext instead of Hom

For categories $\mathcal{C}$ and $\mathcal{D}$, a pair of functors $\mathcal{C} \xrightarrow{\;R\;} \mathcal{D}$ and $\mathcal{C} \xleftarrow{\;L\;} \mathcal{D}\,$ are an adjoint pair if for any ...
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112 views

Is this functor a left adjoint?

I have a functor $sGrph \to Quiv$ which replaces each edge in a given simple graph with a cospan (and leaves the vertices alone). This functor clearly preserves colimits. Is it a left adjoint?
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idempotent in quiver theory

I am studyng quivers and algebras of the text "Elements of the representation theory of associative algebras", author Assem. In the page 46 says: "any idempotent $ε$ of $ε_a(KQ)ε_a$ can be written in ...
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All linear representations of a finite group are quiver representations?

Definitions A linear representation of a finite group $G$ is a group homomorphism: $$\rho:G\to\text{GL}(V),$$ where we shall assume that $V$ is a finite dimensional vector space. We call $V$ the ...
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Meaning of “fully decomposable”

For the quiver $1 \rightarrow 2$, we can consider the category of representations $Rep \;(Q)$. The objects are representations of this quiver, and the morphisms are such that we have $$\require{AMScd}...