All Questions
Tagged with rt.representation-theory quivers
129 questions
27
votes
3
answers
2k
views
How can classifying irreducible representations be a "wild" problem?
Let $q$ be a prime power and $U_n(\mathbb{F}_q)$ be the group of unitriangular $n\times n$-matrices. I've read and heard in several places (see e.g. this mathoverflow question) that classifying ...
- 2,450
3
votes
2
answers
362
views
Do morphisms of finitely-decomposable Quiver representations map indecomposables nicely?
Consider two quivers $Q$ and $Q'$ of type $A_n$, laid out horizontally like so:
Given representations of $Q$ and $Q'$, Gabriel's theorem guarantees the existence of finitely many indecomposables for ...
- 15.5k
5
votes
0
answers
426
views
Homological dimension of completed path algebras.
Let A = c[Q]/I be a finite dimensional quotient of a path algebra over a quiver Q, with I being the ideal of relations.
Is it true that the I-adic completion of A has finite homological dimension?
- 51
7
votes
1
answer
262
views
Description of modules over self-injective algebras of finite representation type
Is there any description of indecomposable modules and irreducible morphisms over self-injective algebras of finite representation type? I am interested mainly in such a description for nonstandard ...
- 71
10
votes
1
answer
648
views
The Fukaya category of a simple singularity (reference request)
I have heard that for an ADE singularity $f$,
$ D^b\mathrm{Fuk}(f) \simeq D^b(\mathrm{Rep}\ Q)$
where $Q$ is the corresponding Dynkin quiver. (As one would hope, if $\mathrm{Fuk}$ is some kind of ...
- 853
4
votes
1
answer
280
views
Vertex embeddings of quantum groups via quivers
Let Uq be a quantised enveloping algebra of type affine ADE (untwisted). By the loop presentation of Uq, we see that for each vertex of the finite Dynkin diagram, there is an inclusion Uq($\hat{sl_2}$)...
- 8,866
6
votes
2
answers
917
views
Do fixed point sets in equivariant crepant resolutions have the same cohomology? How about for the specific case of Nakajima quiver varieties?
A crepant resolution $f:Y\to X$ is a resolution of singularities with $f^*(K_X)=K_Y$. Crepant resolutions do not always exist, and when they exist they may not be unique. However, different crepant ...
- 2,372
13
votes
0
answers
615
views
The derived category of integral representations of a Dynkin quiver
Let $Q$ be a Dynkin quiver. Let $\mathbb CQ$ be its complex path algebra. It is defined in a way such that modules over $\mathbb CQ$ are the same as representations of the quiver $Q$. Let's write $\...
- 3,174
3
votes
3
answers
648
views
Quiver on tensor product
Let $Q=(Q_{0},Q_{1},h,t)$ be a finite quiver where $Q_{0}$ are the vertices, $Q_{1}$ the arrows and we have two maps $h: Q_{1} \rightarrow Q_{0}$ (head) and $t: Q_{1} \rightarrow Q_{0}$ (tail). Fix a ...
- 65
3
votes
1
answer
219
views
Ring completion of $kQ$
Hello,
Let $Q$ be a finite quiver, let $M$ denote the arrow ideal and let $kQ$ denote the path algebra. Endow $kQ$ with the $M$-adic topology. Now let $\mathcal{A}$ be the set of all formal series ${\...
- 65
2
votes
1
answer
281
views
Indecomposable extensions of regular simple modules by preprojectives
Given four points in general position on $\mathbb{P}^2$ there exists a projection to $\mathbb{P}^1$ collapsing these four pairwise to two points. Its kernel is some fifth point on $\mathbb{P}^2$.
In ...
- 853
3
votes
1
answer
845
views
An algorithm for constructing the AR-quiver of a path algebra corresponding to a change in the orientation.
Considering the path algebra of the quiver $\mathbb{A}_n$, it is well known its Auslander-Reiten quiver with the canonical orientation of $\mathbb{A}_n$, that is, with all the arrows from, say, left ...
5
votes
1
answer
513
views
Morita equivalence of acyclic categories
(Crossposted from math.SE.)
Call a category acyclic if only the identity morphisms are invertible and the endomorphism monoid of every object is trivial. Let $C, D$ be two finite acyclic categories. ...
- 118k
8
votes
1
answer
805
views
Quivers of selfinjective algebras.
Let's say a quiver $Q$ is covered by cycles if each of it’s arrows can be included in an oriented cycle.
It's easy to prove that if a path-algebra with relations $KQ/I$ (where $I$ is an admissible ...
- 83
34
votes
8
answers
8k
views
Are quivers useful outside of Representation Theory?
There is a trend, for some people, to study representations of quivers. The setting of the problem is undoubtedly natural, but representations of quivers are present in the literature for already >...
- 1,437
5
votes
1
answer
1k
views
Indecomposable modules over preprojective algebras
Would you please give some references concerning the number of indecomposable modules over preprojective algebras of type $A_n$?
More precisely, I need references about the following claim: The ...
- 3,078
32
votes
3
answers
5k
views
Why did Gabriel invent the term "quiver"?
A quiver in representation theory is what is called in most other areas a directed graph. Does anybody know why Gabriel felt that a new name was needed for this object? I am more interested in why he ...
4
votes
1
answer
382
views
Irreducibility of the F-polynomial of an indecomposable quiver representation
Question. Is the $F$-polynomial of an indecomposable quiver representation irreducible?
Here the $F$-polynomial is the generating function of the Euler characteristics of quiver Grassmannians, that ...
13
votes
0
answers
563
views
Are the extra vertices in Nakajima's doubling of a quiver related to Langlands duality?
To define a Nakajima quiver variety associated to a quiver $Q = (Q_0,Q_1)$
(vertices and arrows), one first doubles it to $Q^\heartsuit$ by attaching
an extra vertex to every old vertex in $Q_0$. Then ...
- 27.8k
6
votes
1
answer
277
views
A question about saturation of quivers
Let $Q$ be an acylic quiver. Let $E$ and $F$ be finite dimensional representations, with $E$ indecomposable. Suppose that, for some positive integer $r$, the representation $F$ injects into $E^{\oplus ...
- 156k
1
vote
1
answer
268
views
Hereditary algebras as quotient algebras
This is the first time I post a question on MO, so I'm shy a liite bit. Can you give a "non-trivial" example of a finite dimensional hereditary algebra which is quotient of an infinite dimensional ...
- 13
10
votes
3
answers
1k
views
Are the underlying undirected graphs of two mutation-equivalent acylic quivers isomorphic?
Quiver mutation, defined by Fomin and Zelevinsky, is a combinatorial process. It is important in the representation theory of quivers, in the theory of cluster algebras, and in physics.
We consider ...
- 101
15
votes
2
answers
3k
views
when are algebras quiver algebras ?
Good Morning from Belgium,
I'm no stranger to the mantra that quiver-algebras are an extremely powerful tool (see for example the representation theory of finite dimensional algebras). But what is a ...
12
votes
3
answers
1k
views
construct scheme from quivers?
I heard from some guys working in noncommutative geometry talking about the idea that one can construct the noncommutative space from quivers. I feel it is rather interesting. However, I can not image ...
- 1,305
11
votes
1
answer
792
views
What's known about the stalks of Lusztig's perverse sheaves on quiver varieties?
Lusztig has defined a category of perverse sheaves on the moduli space of representations of a Dynkin quiver (see his paper) corresponding to canonical basis vectors.
I'm interested in the stalks ...
- 44.7k
5
votes
3
answers
781
views
Acyclic quivers differing only in arrow directions: functorial isomorphism of representation categories?
Let $Q$ and $R$ be two acyclic quivers which differ only in the directions of their arrows (i. e., the underlying undirected graphs are the same).
1. Does there exist an isomorphism of additive ...
- 33.8k
11
votes
2
answers
2k
views
What do the local systems in Lusztig's perverse sheaves on quiver varieties look like?
In "Quivers, perverse sheaves and quantized enveloping algebras," Lusztig defines a category of perverse sheaves on the moduli stack of representations of a quiver. These perverse sheaves are defined ...
- 44.7k
8
votes
1
answer
1k
views
Global dimenson of quivers with relations
Let Q be a finite quiver without loops. Then its global dimension is 1 if it contains at least one arrow.
I'm trying to get some intuition about how much the global dimension can grow when we ...
- 10.7k
18
votes
7
answers
2k
views
ubiquity, importance of path algebras
I work in planar algebras and subfactors, where the idea of path algebras on a graph (alternately known as graph algebras, graph planar algebras, etc.) is quite useful. The particular result I'm ...
- 1,089