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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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 ...
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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, ...
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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 ...
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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 ...
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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 ...
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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 ...
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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 ...
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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 ...
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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 ...
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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?
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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 $D$...
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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 ...
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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 ...
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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 ...
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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 ...
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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)....
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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 ...
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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 $G$, $G'$ ...
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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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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 ...
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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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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 $x^{p}=x^p$. ...
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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 &...
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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 ...
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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 ...
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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 $Q=(Q_0,Q_1)$, ...
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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 ...
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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 $...
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Example of a brick-infinite, tame, triangular algebra of global dimension$\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 ...
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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\...
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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 ...
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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 ...
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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 ...
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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-\...
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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 \...
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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-...
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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 ...
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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 ...
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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 ...
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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 ...
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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 ...
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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 ...
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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 $\...
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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....
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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 ...
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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-...
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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: $\forall m,n,n'\in M:0\neq m*n =m*n' \Rightarrow n=n'$.
...
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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 ...
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Finite-dimensional irreducible representations of 2-loop quiver
What are the finite-dimensional irreducible representations of the quiver with one vertex and two loops?
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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 ...
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