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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.

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Ringel's interpretation of quantum groups as Hall algebras at $q=1$

Let $Q$ be a finite-type quiver and let $\mathfrak{g}$ be the semisimple Lie algebra associated with the corresponding simply-laced Dynkin diagram. Let $U_v^+(\mathfrak{g})$ be the positive part of ...
Antoine Labelle's user avatar
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Hall algebra of constructible functions of affine quiver?

I have read in "Quiver Representations and Quiver Varieties" by Kirillov that Hall algebra of constructible functions are defined only for Dynkin quivers because they are of finite type. So ...
user236626's user avatar
6 votes
1 answer
139 views

Quiver variety, generically symplectic

Theorem 11.3.1 (iv) of "Noncommutative Geometry and Quiver algebras" by Crawley-Boevey, Etingof and Ginzburg claims that, for a dimension vector $\mathbf{d}\in\Sigma_0$, the quiver variety $...
Qwert Otto's user avatar
2 votes
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72 views

Does this finitely-generated algebra have a name?

I've been led to consider certain finitely-generated algebras that arise from some Coxeter groups (finite and affine Weyl groups at least). As a very concrete example, consider the infinite dihedral ...
Marty's user avatar
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Does every SES of injective bounded cochain complexes split?

Question: Does every short exact sequence of injective bounded cochain complexes, $0\rightarrow I^\bullet\rightarrow J^\bullet\rightarrow K^\bullet\rightarrow 0$, split? I am interested in a discrete ...
Ondrej Draganov's user avatar
1 vote
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Multiplicative bases, path algebras, and Ext algebras

I am interested in understanding when a multiplicative basis exists for finite dimensional algebras over an algebraically closed field, and, in particular, Ext-algebras that are finite dimensional. It ...
James Steele's user avatar
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0 answers
103 views

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 ...
It'sMe's user avatar
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Projective resolution of a quiver with relations

How do we compute the projective resolution of a representation of a quiver with relations. For example consider the Beilinson quiver $B_4$ $. with the relations ­$\{\alpha_j^k\alpha_i^{k-1}=\alpha_i^...
user52991's user avatar
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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 ...
It'sMe's user avatar
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Auslander-Reiten sequences where irreducible morphisms are all epi/mono

Let's work in the setting of modules over an Artin algebra $A$, or a finite-dimensional $k$-algebra $A$, or if you like, modules over a connected quiver $Q$ without oriented cycles. Let $M$ be such a ...
Marty's user avatar
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4 votes
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Proof of McKay correspondence without classifications

$\DeclareMathOperator\SU{SU}$I am wondering if there is any known proof of the McKay correspondence (I will give the precise statement that I mean by this) that doesn't use the classification of ...
Max Steinberg's user avatar
9 votes
1 answer
236 views

Formal smoothness of path algebras and connections

Let $k$ be a field of characteristic zero and $A = kQ$ the path algebra associated with a quiver $Q$. The algebra $A$ is said to be formally smooth over $k$ if $$ \Omega^1_kA = \operatorname{Ker}(\...
Qwert Otto's user avatar
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0 answers
70 views

geometric objects in quiver variety corresponding to short exact sequences

I was studying quiver variety and known that representations of a quiver correspond to points in the corresponding quiver variety. So if give you a fixed triple representations $(M_1,M_2,M_3)$, I was ...
user236626's user avatar
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71 views

"Approximating" ring of semi-invariants

I'm trying to calculate the semi-invariant ring for certain types of quivers. For a very brief introduction to semi-invariant rings of quiver please have a look at this wikipedia article at the ...
It'sMe's user avatar
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257 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 ...
It'sMe's user avatar
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150 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, ...
Andrea B.'s user avatar
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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 ...
Benjamin Steinberg's user avatar
2 votes
1 answer
298 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 ...
Joshua Grochow's user avatar
2 votes
0 answers
86 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 ...
It'sMe's user avatar
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2 votes
1 answer
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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 ...
It'sMe's user avatar
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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 ...
OOOOOO's user avatar
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123 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 $D$...
wlad's user avatar
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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 ...
It'sMe's user avatar
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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 ...
Mare's user avatar
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1 vote
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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 ...
It'sMe's user avatar
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143 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 ...
Laurent Cote's user avatar
1 vote
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208 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)....
It'sMe's user avatar
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8 votes
3 answers
431 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 ...
darij grinberg's user avatar
2 votes
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129 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 $G$, $G'$ ...
It'sMe's user avatar
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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 ...
It'sMe's user avatar
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2 votes
1 answer
186 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 ...
It'sMe's user avatar
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127 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 ...
It'sMe's user avatar
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2 votes
0 answers
62 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 $x^{p}=x^p$. ...
Mare's user avatar
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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 &...
IMP's user avatar
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4 votes
0 answers
259 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 ...
It'sMe's user avatar
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3 votes
0 answers
111 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 $Q=(Q_0,Q_1)$, ...
It'sMe's user avatar
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2 votes
0 answers
102 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 ...
tox123's user avatar
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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 $...
It'sMe's user avatar
  • 839
1 vote
0 answers
128 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 ...
It'sMe's user avatar
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1 vote
0 answers
115 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\...
It'sMe's user avatar
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2 votes
0 answers
91 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 ...
Mare's user avatar
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2 votes
0 answers
268 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 ...
It'sMe's user avatar
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0 votes
1 answer
265 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 ...
It'sMe's user avatar
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1 vote
1 answer
218 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-\...
It'sMe's user avatar
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5 votes
3 answers
812 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 \...
Piotr Pstrągowski's user avatar
6 votes
1 answer
312 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-...
It'sMe's user avatar
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4 votes
1 answer
108 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 ...
Richard Chen's user avatar
1 vote
0 answers
98 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 ...
Subham Jaiswal's user avatar
1 vote
0 answers
144 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 ...
It'sMe's user avatar
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2 votes
2 answers
140 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 ...
Sergey Guminov's user avatar