# 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

1
vote

0
answers

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

0
answers

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, ...

3
votes

0
answers

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 ...

5
votes

0
answers

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 ...

0
votes

0
answers

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 ...

2
votes

1
answer

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 ...

2
votes

0
answers

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 ...

2
votes

1
answer

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

0
answers

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 ...

0
votes

0
answers

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

0
answers

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

0
answers

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 ...

2
votes

0
answers

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

0
answers

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 ...

2
votes

0
answers

112
views

### Two notions of stability

Let $Q$ be a finite quiver (i.e. an oriented graph). A representation of $Q$ is by definition a module over the path algebra of $Q$. More concretely, a representation associates to every vertex $v \in ...

1
vote

0
answers

206
views

### Is $\langle\chi,\lambda\rangle=0$, whenever the limit exists? Where is the mistake?

Suppose $G$ is a linearly reductive algebraic group acting linearly on a finite dimensional vector space $V$ over $\mathbb{C}$. This induces an action on the coordinate ring $\mathbb{C}[V]$ (see here)....

7
votes

3
answers

394
views

### Smallest faithful representation of an upper-triangular matrix quotient

This is a curiosity question that came out of teaching abstract algebra.
Let $F$ be a field, and $n>1$ an integer.
Let $F^{n \leq n}$ be the $F$-algebra of all upper-triangular $n\times n$-matrices ...

2
votes

0
answers

109
views

### Getting an equivariant morphism

Let $X\subset\mathbb{A}^n$ be an irreducible affine variety over an algebraically closed field $\mathbb{K}$ of characteristic zero. Suppose we have two linearly reductive algebraic groups $G$, $G'$ ...

1
vote

0
answers

52
views

### 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 ...

2
votes

1
answer

162
views

### Orbits in the open set given by Rosenlicht's result

Let $G$ be a linearly reductive algebraic group, and let $X$ be an irreducible affine variety, over an algebraically closed field $\mathbb{K}$, with a regular action of $G$. By Rosenlicht's result, we ...

1
vote

0
answers

122
views

### 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 ...

2
votes

0
answers

55
views

### Quiver and relations for Hopf algebras associated to quiver algebras

Let $A=KQ/I$ be a finite dimensional quiver algebra with admissible relations $I$.
$A$ can be made into a restricted Lie algebra over a field of characteristic $p$ via
$[x,y]=xy-yx$ and $x^{p}=x^p$. ...

3
votes

1
answer

102
views

### References on coefficient quivers

I would like to study about coefficient quivers, but I cannot find a good reference, as book for example. I could find many papers working with coefficient quivers, but none of them give a book or a &...

4
votes

0
answers

224
views

### Road map for learning cluster algebras

I'm a PhD student and I would like learn about cluster algebras. I'm wondering what is a good reference (i.e., has detailed explanations, examples, etc) to learn from the basic and what are some of ...

0
votes

0
answers

65
views

### How to canonically induce a morphism between module categories of path algebras when given a morphism of quivers?

Let $\pi:P\to Q$ be a morphism of quivers, i.e., $s(\pi(a))=\pi(s(a)),t(\pi(a))=\pi(t(a)) $ for arrows $a$, start $s$ and tail $t$. Is there a canonical way to induce from $\pi$ a morphism between ...

3
votes

0
answers

97
views

### 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 $Q=(Q_0,Q_1)$, ...

2
votes

0
answers

98
views

### When do two path algebras share an underlying graph?

Suppose $Q$ and $Q'$ are two quivers. I am curious as to what relation $\mathbb{C}Q$ bears to $\mathbb{C}Q'$ when $Q$ and $Q'$ share the same underlying graph and only differ by direction.
Since ...

1
vote

0
answers

95
views

### Is $U\subseteq X^{s}(L)$?

Let $X$ be an irreducible affine variety over an algebraically closed field $\mathbb{K}$ of characteristics $0$. Let $G$ be a connected, linearly reductive, affine algebraic group acting regularly on $...

1
vote

0
answers

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

0
answers

108
views

### 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\...

2
votes

0
answers

78
views

### Quiver and relations for group algebras of p-groups

Let $G$ be a finite $p$-group and $K$ a field of characteristic $p$.
$KG$ is isomorphic to a quiver algebra $KQ/I$ with admissible ideal $I$.
Question 1: Does there always exist such an $I$ where the ...

2
votes

0
answers

232
views

### Understanding a proof of a result of Schofield

I'm reading a paper of Aidan Schofield- "General Representations of Quivers" and I'm trying to understand the proof of Theorem 3.3. I'm having trouble understanding the argument that's ...

0
votes

1
answer

206
views

### Quiver representations over any commutative ring

I'm reading a paper of Aidan Schofield "General Representations of Quivers" and he defines quiver representation over any commutative ring. See the below image.
Towards the end, he has this ...

1
vote

1
answer

205
views

### A result of Schofield in the case of quivers with relations

Let $Q$ be a quiver without oriented cycles. A result of Schofield says that, for dimension vectors $\alpha$ and $\beta$ of $Q$, $\beta\hookrightarrow\alpha$ iff $\operatorname{ext}(\beta, \alpha-\...

5
votes

3
answers

764
views

### Quiver with two objects and two arrows composing to zero

In the description of the integral Adams spectral sequence, representations of the following quiver (with relations) arise naturally:
We have two objects $A, B$,
we have two arrows $\pi: A \...

6
votes

1
answer

297
views

### Prove that $\overline{a}_{11}$ is a prime element in $R$

Consider the affine space given by four $2\times 2$ matrices, i.e., $\mathbb{A}^{16}\cong M(\mathbb{C})_{2\times 2}^4$. Now, consider the algebraic set $V$ given by the vanishing of the relation $AB-...

4
votes

1
answer

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

0
answers

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

0
answers

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 ...

2
votes

2
answers

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 ...

3
votes

1
answer

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 ...

2
votes

0
answers

104
views

### Potential on a quiver

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 ...

0
votes

1
answer

248
views

### Quiver varieties associated to D_4

Let $Q=(I,\Omega)$ be the $D_4$ affine quiver. We choose as dimension vector $(2,1,1,1,1)$ (where $2$ is on the central vertex). As this dimension vector is indivisible, we can choose a generic $\...

14
votes

2
answers

930
views

### Invariants of matrices (by simultaneous $\mathrm{GL}_n$ conjugation) over arbitrary rings

$\DeclareMathOperator\GL{GL}$Let $R$ be a commutative ring, let $R[n] := R[M_d^{\oplus n}]$ be the polynomial ring on $nd^2$ variables corresponding to the coordinates of $n$-many $d\times d$ matrices....

4
votes

2
answers

811
views

### Research topics in representation theory of algebras [closed]

I was wondering what are some of the hot topics in quiver representation or representation theory of algebras that can lead to good mathematics and is important to many mathematicians and top ...

5
votes

0
answers

344
views

### What representation theoretic properties does the semi-invariant ring tell us?

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-...

1
vote

0
answers

168
views

### What is this algebraic object (special case of a semigroup)?

Let $(M,*)$ be a finite semigroup. Further we demand the following:
Zero element: $\exists0\in M \forall m\in M:0*m=0=m*0$.
Left cancelation: $\forall m,n,n'\in M:0\neq m*n =m*n' \Rightarrow n=n'$.
...

3
votes

0
answers

168
views

### A conceptual explanation for the Kirchoff matrix theorem in terms of the quiver algebra

On the wikipedia page for the Kirchoff matrix theorem, they state a souped up version of the theorem:
Let $G$ be a finite undirected loopless graph and let us form the square matrix $L$ indexed by the ...

5
votes

0
answers

210
views

### Finite-dimensional irreducible representations of 2-loop quiver

What are the finite-dimensional irreducible representations of the quiver with one vertex and two loops?

6
votes

2
answers

451
views

### Tensor of finite-dimensional algebra over perfect field is semisimple

Let $K$ be a field and let $\Lambda_{1}$ and $\Lambda_{2}$ be two finite-dimensional $K$-algebras with Jacobson radicals $J_{1}$ and $J_{2}$ respectively. How to show or where can I find the proof of ...