# Questions tagged [quivers]

"Quiver" is the word used for "directed graph" in some parts of representation theory. The main reason to use the term quiver is to indicate an interest in considering representations of the quiver.

184 questions
Filter by
Sorted by
Tagged with
1 vote
215 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 ...
1 vote
113 views

### Number of cluster variables associated to A type quivers

In a seminar/reading course about cluster algebras, we came across the fact that the number of cluster variables for the cluster algebra associated to mutating the quiver $A_n$ is $n(n+3)/2$ (rather, ...
95 views

### Mutations in triangulated category and cluster algebra

Let $\mathcal{D}$ be an enhanced triangulated category (basically meaning that $\operatorname{Hom}$'s are complexes). There is the notion of mutation in an enhanced triangulated category: given a full ...
93 views

### Modern proof of a theorem of Dickson on finite representation type

In Theorem 3.1 the paper S. Dickson, On algebras of finite representation type Trans. Amer. Math. Soc. 135 (1969), 127-141, Dickson gives a sufficient condition for an algebra to have infinite ...
86 views

### Connected components of $Q(\mathrm{s\tau-tilt}A)$

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 algebraically closed field, which is ...
157 views

### Tame/wild classification of *cyclic* quivers?

There is a famous classification of the path algebras of finite acyclic quivers into finite, tame, and wild representation types. For quivers with cycles, it is standard that the 2-loop quiver (with ...
79 views

### Example of a triangular string algebra that is rep infinite, but $\tau$-tilting finite

Let $A$ be a finite dimensional algebra over an algebraically closed field $\mathbb{K}$. Hence, $A$ can be realized as the path algebra of a bound quiver $(Q,I)$, where $I\subseteq\mathbb{K}Q$ is an ...
126 views

### Rep infinite, but $\tau$-tilting finite

Let $A$ be a finite dimensional algebra over an algebraically closed field. I'm trying to better understand the difference between $A$ being representation infinite and $A$ being $\tau$-tilting ...
1 vote
52 views

### When the dg cluster category of a quiver is saturated?

Let $Q$ be a finite quiver without oriented cycles. In https://arxiv.org/abs/0807.1960 , Keller defines the dg cluster category $C_Q$ of $Q$. When is $C_Q$ smooth proper dg-category? If $Q$ is a ...
48 views

### Rank of Coxeter matrix

Let $Q$ be a quiver and $\Phi_Q$ be the Coxeter matrix of $Q$. Then $\Phi_Q\pm I$ are full-rank?
1 vote
114 views

### Quiver representations and the standard matrix decompositions

Many matrix decompositions - like the Jordan Normal Form, the SVD, the spectral theorem, the Takagi decomposition - have the property that they express a matrix $M$ as the form: $$M = A D B$$ where $D$...
1 vote
56 views

### Structure of tame concealed algebra of Euclidean type

I wanted to know some references where people have studied the representation theory of tame concealed algebra of Euclidean type. What do we know about the structure of their module category? What ...
56 views

### Number of admissible quotient algebras

Let $Q$ be a finite connected quiver. An admissible quotient algebra is an algebra of the form $KQ/I$ with an admissible ideal $I$. Question 1: Is there a nice closed formula for the number of ...
1 vote
77 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 ...
112 views

1 vote
123 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 ...
1 vote
108 views

764 views

94 views

### How much is known about the non-degeneracy of Quiver-with-potential associated to closed punctured surfaces?

The potential of the quiver associated to surfaces is the canonical one given by Labardini-Fragoso's 2009 paper, who proved that the the QP associated to surfaces whose boundary is nonempty is rigid ...
1 vote
74 views

### A sufficient condition for automorphism of an exact sequence

I asked A sufficient condition for Automorphism of an exact sequence earlier on Math.StackExchange but did not get any response so am posting it here. I am given the following commutative diagram with ...
1 vote
131 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 ...
135 views

### Finding exceptional regular representations of $\tilde{D}_4$ efficiently

Let $A$ be the path algebra of the quiver $\tilde{D}_4$. I would like to find its exceptional regular representations with as little computation as possible. Of course, we can compute the whole ...
137 views

### Auslander-Reiten quiver of quiver algebra kQ where Q is of extended dynkin type D4~

I am looking for literature about the Auslander-Reiten quiver of the quiver algebra $kQ$, where $Q$ is of extended dynkin type $\tilde{D_4}$ and $k$ is an algebraically closed field. Does somebody ...
I found two definitions of potential on a quiver. Selfinjective quivers with potential and 2-representation-finite algebras, Martin Herschend and Osamu Iyama 2.1 Quivers with potential. Let $Q$ be a ...