# 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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### 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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### 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 ...
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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 a non-hereditary algebra whose $g$-vector fan is not complete

I'm reading the paper of Pierre-Guy Plamondon, Toshiya Yurikusa and Bernhard Keller "Tame Algebras Have Dense $g$-Vector Fans". I wanted to know some explicit examples of non-hereditary ...
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1 vote
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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 ...
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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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### Is the connected assumption necessary in the following theorem?

I'm reading about the following facts from the book 'Elements of the Representation Theory of Associative Algebras' by Assem, Simpson, and Skrowronski. Note that the algebras involved are not assumed ...
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### How to read off distinguished triangles and cluster-tilting objects in the cluster category of a Dynkin quiver?

I'm new to triangulated category and tilting theory. To illustrate, in $Q=A_4$ the module $M=kQ$ is cluster-tilting. While I know that $M$ satisfies $\mathrm{Ext}(M,M)=\mathrm{Hom}(M,M)=0$ by some ...
1 vote
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### A doubt in proof of existence of Auslander Reiten Sequences

I am providing the context. You can skip to the actual doubt below. My doubt doesnt need knowledge of quivers or even almost split sequences. I am reading Theorem 5.4.10 from the book An introduction ...
1 vote
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### An application of Snake Lemma

I am stuck at undderstanding a claim made in Derksen's book An Introduction to Quiver Representations. This is Lemma 2.4.3. But the part I am stuck at is purely homological algebra it seems. Let $A$ ...
1 vote
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### Non-empty stable locus in 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 us ...
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### Termination of Buchberger's algorithm for path algebras

I'm reading the paper Noncommutative Gröbner Bases, and Projective Resolutions by Edward L. Green, which presents a version of Buchberger's algorithm for path algebras. I'm trying to show that the ...
1 vote
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### Dimensions of homomorphisms determines the module

I am reading this paper about degenerations for representations. I am mentioning some context though all of it will not be needed I think for my question. We basically have a finite dimensional ...
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### Viewing Path Algebras as Matrix algebras

Given a Path Algebra $\mathbb{C}Q$ of an acyclic directed graph(also called a Quiver in this theory) $Q$. I am interested in finding a corresponding matrix algebra for it if possible. I am able to do ...
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### no. of minimum generating sets of $M_{n}(\mathbb{C})$

Problem: How many $n \times n$ pairs $(A, B)$ of (0 , 1) matrices are there which generate the whole algebra $M_{n \times n}(\mathbb{\mathbb{C}})$? I don't know if this question is open or hopelessly ...
1 vote
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### Cluster algebras from quivers vs. cluster algebras from skew-symmetrizable matrices

In a course on cluster algebras we first defined the cluster algebra arising from a quiver. Next we saw that each quiver gives rise to an exchange matrix that is skew-symmetric. So we generalized to ...
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### Derived functor of a reflection (BGP) functor

I'm studying the book "quiver representations and quiver varieties" of Kirillov and I'm in Theorem 3.10: The functor $\Phi_i^+$ is left exact. Moreover, $R^n\Phi_i^+(V)=0$ for all $n>1$, ...
1 vote
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### What if the semi-invariant ring is a polynomial ring or hypersurface

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-...
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### Why do people study semi-invariant ring (in general)?

I'm interested in studying about semi-invariant ring in the context of Quiver representations. I started reading about it in some books and few places on the internet. But, nowhere do they mention why ...
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### What is the quiver representation of an algebra, and how is it obtained from a decomposition into spaces of morphisms (the opposite algebra)?

I will make everything concrete so I have something to work with. Let $A$ be an algebra, free, and 4-dimensional, and let $A = P_1 \oplus P_2$ be a decomposition into two 2-dimensional free modules. I ...
Need some help with Gabriel's theorem, doing the part "if a graph is of finite type, it must be a Dynkin graph" like this: Let $\vec{Q}$ be of finite type. All representations is a direct ...
### Intuitive explanation of quiver $Q^{(n,s)}$.
On page 6 of this article, the author defined a new quiver. Fix positive integers $n$ and $s$. We define the quiver $Q:=Q^{(n, s)}$ with the set $Q_{0}$ of vertices and the set $Q_{1}$ of arrows by  ...