# 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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### 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 ...
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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 ...
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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 ...
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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 ...
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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 ...
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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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