Questions tagged [cluster-algebras]
Questions related to cluster algebras, a class of commutative rings introduced around 2000 by Fomin and Zelevinsky, and nearby topics.
104 questions
2
votes
1
answer
223
views
Cluster algebra structure on the coordinate ring of $Mat_3$
Let $Mat_3$ be the set of all 3 by 3 matrices. I have some questions on the cluster algebra structure on the coordinate ring of $Mat_3$.
We use $\Delta_{j_1\ldots j_n}^{i_1\ldots i_n}$ to denote the ...
- 1,238
1
vote
1
answer
216
views
Why do finitely many cluster variables imply finitely many y-variables?
Suppose we have a seed $(x,y,B)$ where $B$ is a skew-symmetrizable matrix, $x = \{x_1,\ldots,x_n\}$ and y is an n-tuple of elements in $Trop\{x_{n+1},\ldots,x_m\}$.
If there are finitely many cluster ...
- 31
4
votes
1
answer
119
views
Rigid regular objects of path algebras of tame quivers
In the paper On Maximal Green Sequences by Brustle, Dupont and Perotin the authors argued that in a path algebra $\Lambda=kQ$ of a tame quiver $Q$ with $n$ vertices each tilting module contains at ...
CommunityBot
- 1
1
vote
0
answers
168
views
(b,c) rank 2 cluster algebras
Let $x$ and $y$ be variables. Consider the following recurrence:
\begin{equation}
u_{n}:=
\begin{cases}
\displaystyle{\frac{1+u_{n-1}^b}{u_{n-2}}} & if\ n\ \text{is even},\\
&\\
\displaystyle{\...
- 307
0
votes
0
answers
195
views
Automorphisms of weighted quiver
I am reading this paper strongly primitiv species with potentials I : mutations.
In page 6, they give the definition of weighted quiver: a weighted quiver is a pair $(Q,d)$, where $Q$ is a loop-...
- 775
4
votes
1
answer
131
views
Geometric realizations of cluster categories of non-simply-laced types
Geometric realizations of cluster categories of simply-laced types are studied in the following papers.
Philippe Caldero, Frédéric Chapoton, and Ralf. Schiffler, Quivers with relations arising from ...
- 156k
6
votes
1
answer
300
views
What is the status of a problem about cluster categories?
Let $H$ be a hereditary algebra of Dynkin type. There is a cluster category $\mathcal{C}_H$ defined by Aslak Bakke Buan, Robert Marsh, Markus Reineke, Idun Reiten, and Gordana Todorov in Tilting ...
- 1,690
15
votes
1
answer
2k
views
Applications of cluster algebras
Why are so many algebraists nowadays interested in cluster algebras?
(This is a rewording of one half of the closed question Cluster algebras and teichmuller theory.)
- 1,690
9
votes
2
answers
978
views
A question about the quivers with potentials
Let $Q=(Q_0,Q_1,h,t)$ be a quiver consisted of a pair of finite sets $Q_0$(vectors),and $Q_1$ (arrows) supplied with two maps $h : Q_1 → Q_0$ (head) and $t : Q_1 → Q_0$ (tail ). This definition allows ...
14
votes
1
answer
568
views
Is there some relation between cluster algebras and crystal graphs?
Cluster algebras are closely related to totally positivity in algebraic groups and canonical bases in quantum groups.
Is there some relation between cluster algebras and crystal graphs? Can the ...
- 2,516
2
votes
0
answers
73
views
Question about the mutation of a cluster seed associated to any word of the braid semigroup
Let $G$ be a semisimple Lie group with the set of positive simple roots $\prod$. Let $\prod^{-}$ be the set of negative simple roots and let $\mathfrak{M}$ be the semigroup freely generated by $\prod$ ...
- 348
2
votes
0
answers
137
views
Questions about cluster $\mathcal{X}$-varieties and amalgamation
I am trying to learn the amalgamation of two cluster seeds and I am reading the paper https://arxiv.org/pdf/math/0508408.pdf written by V.V. Fock and A. B. Goncharov. I am at a loss for the Lemma 2....
- 348
6
votes
1
answer
137
views
Decidability and Cluster algebras
Recall the definition of a cluster algebra,
which can be seen as a (possibly infinite) graph, where each vertex is a tuple of a quiver and Laurent expressions at some of the vertices of the quiver. ...
- 7,875
2
votes
0
answers
133
views
Weighted quiver in Keller's Java Quiver mutation [closed]
I need to do mutation using weighted quivers. There are weights on the arrows which looks like:
How to draw weighted quiver in Keller's Java Quiver mutation:
https://webusers.imj-prg.fr/~bernhard....
- 6,211
3
votes
1
answer
1k
views
What are the relations among canonical basis, dual canonical basis, Semicanonical Basis, dual semicanonical bases?
I am reading the lecture notes and would like to know more about canonical basis.
What are the relations among canonical basis, dual canonical basis, Semicanonical Basis, dual semicanonical bases?
...
- 7,875
2
votes
1
answer
176
views
Which cluster algebras where the existence of maximal green sequences is still unknown?
Maximal green sequences are studied in many papers. For example, Maximal Green Sequences for Cluster Algebras Associated to the n-Torus by Eric Bucher, On Maximal Green Sequences by Thomas Brüstle, ...
- 7,875
3
votes
2
answers
234
views
How to translate cluster X-coordinates to cluster A-coordinates?
In the paper, (5.28) on page 35 is a formula which translate cluster A-coordinates to cluster X-coordinates:
\begin{align}
x_i = \prod_{i \to j} a_j.
\end{align}
Is there a formula which translate ...
- 6,211
4
votes
2
answers
373
views
Reference request: Associahedron
I am reading Sergey Fomin's and Nathan Reading's paper Root Systems and Generalized Associahedra.
I need a good reference for associahedron of classical types. Besides, whether there are some ...
- 6,302
2
votes
0
answers
143
views
Shear coordinates, lambda lengths, cluster variables
I am trying to understand the relations among Shear coordinates, lambda lengths, cluster variables, in the paper. Is the following correct?
Lambda lengths = cluster A-variables
Shear coordinates = ...
- 6,211
0
votes
2
answers
167
views
Tropical version of exchange relations in cluster algebras
The exchange relation in a cluster algebra is
\begin{align}
x_k' = \frac{1}{x_k} (\prod_{j \to k}x_j + \prod_{k \to j} x_j).
\end{align}
Do we have some tropical version of this relation? Are there ...
- 6,211
4
votes
2
answers
243
views
References about tropical cluster algebras and tropical Laurent phenomenon
Are there some references about tropical cluster algebras and tropical Laurent phenomenon? I searched on Google but only found one paper: Tropical Plucker functions and their bases
.
Thank you very ...
- 156k
3
votes
1
answer
313
views
Trying to understand the proof of Laurent phenomenon of cluster algebras
I am trying to understand the proof of Laurent phenomenon of cluster algebras in the book (Sergey Fomin, Lauren Williams, Andrei Zelevinsky, Introduction to Cluster Algebras. Chapters 1-3, arXiv:1608....
- 33.8k
0
votes
1
answer
204
views
Mutation equivalence of quivers
Given two orientations $Q, Q'$ of a Dyinkin diagram. Is it always true that after a sequence of mutations, $Q$ becomes $Q'$? Are the some references about this? Thank you very much.
0
votes
1
answer
241
views
Canonical basis of cluster algebras
Let $x_{k+1} = \frac{x_k^{d_k}+1}{x_{k-1}}$, $k \in \mathbb{Z}$, where $d_{k+2} = d_k \in \mathbb{Z_{>0}}$. Let $b=d_1$ and $c=d_2$. Define the cluster algebra $A = A(\left( \begin{matrix} 0 & ...
- 7,875
1
vote
1
answer
72
views
A question about exchange pattern
Exchange pattern, see Section 2 in "cluster algebras I: foundations" by Fomin and Zelevinsky or How to understand exchange pattern?
Given an example $\cdots \overset{2}{-} t_1 \overset{1}{-} t_2 \...
0
votes
1
answer
262
views
What are the cluster algebra structures on $Gr(3,5)$?
In the paper, cluster algebra structures on $Gr(2,n)$, $Gr(3,6)$, $Gr(3,7)$, $Gr(3,8)$, $Gr(4,6)$ are described. But what are the cluster algebra structures on $Gr(3,5)$ (and $Gr(3,4)$)? Do we have ...
- 2,218
3
votes
1
answer
168
views
How to understand exchange pattern?
I am reading an paper "cluster algebras I: foundations" by Fomin and Zelevinsky.
Let $I = \{1,2, \ldots, n\}$ and $\mathbf{x}$ a cluster.
For each $t \in \mathbb{T}_n$, let $\mathbf{x}(t) = (x_i(t))...
- 7,875
6
votes
1
answer
584
views
Proof of Laurent Phenomenon for Cluster Algebras
I went through the proof of the Laurent phenomenon for Cluster Algebras in Fomin and Zelevinsky's initial paper: Cluster Algebras I: Foundations. I am stuck at their claim that the gcd of two exchange ...
- 7,875
1
vote
1
answer
118
views
Softwares which compute all non-isomorphic quivers in a mutation class
Let $Q$ be a quiver. The mutation class of $Q$ consists of all quivers which can be obtained from $Q$ by a sequence of mutations. Are there some softwares which compute all non-isomorphic quivers in a ...
- 7,875
1
vote
1
answer
216
views
Mutation of valued quivers
Mutations of valued quivers are defined in cluster algebras II, Proposition 8.1 on page 28. I have a question about the number $c'$. For example, let $a = 2, b=1, c=1$ and consider the quiver $Q$:
$1 ...
- 7,875
4
votes
0
answers
239
views
polynomials satisfying the Plücker relation
Let $S_{12}$, $S_{13}$, $S_{14}$, $S_{23}$, $S_{24}$, $S_{34}$ be complex homogeneous polynomials in 4 variables satisfying the Plücker relation :
$$S_{12}S_{34}-S_{13}S_{24}+S_{14}S_{23}=0 .$$
...
- 3,597
1
vote
1
answer
105
views
How to draw a quiver for a pseudoline arragement?
In the lecture notes, on page 24, there is an example of drawing a quiver for a pseudoline arragement. What is the rule to draw a quiver for a pseudoline arragement? I don't know how to put the ...
- 7,875
2
votes
1
answer
315
views
Cluster algebras of finite type
In the webpage, there is a result:
Theorem 1. Coefficient free cluster algebras without frozen variables are in bijection with Dynkin diagrams of type $A_n$, $B_n$, $C_n$, $D_n$, $E_6, E_7, E_8$, $...
CommunityBot
- 1
4
votes
1
answer
212
views
Cluster algebra structure compatible with Poisson brackets
Let $X$ be a Poisson variety. There is a concept "cluster algebra structure compatible with Poisson structure" introduced in the paper.
Suppose that we construct a maximal independent set of ...
- 6,211
4
votes
3
answers
227
views
Cluster Variables for non-convex n-gons
Most of the lectures and lecture notes on Cluster Algebras (at least from Combinatorial point of view) start with mutations of the diagonals of a convex n-gon (mostly the pentagon) as the illustration ...
- 71
3
votes
1
answer
267
views
Quiver folding and maximal green sequences
The technique of quiver folding (please see Folding by Automorphisms) can be used to prove statements about non-simply laced quivers (i.e. valued quivers) when they are already known in the simply-...
- 6,302
2
votes
0
answers
86
views
Characteristics of $c$-vectors of acyclic cluster algebras
In Speyer and Thomas's work, Acyclic Cluster Algebras Revisited the characteristics of $c$-vectors of cluster algebras with the $B$-matrix of the initial seed acyclic are given in Theorem 1.4. Do we ...
- 417
1
vote
1
answer
172
views
Why are exchange graphs of quivers with the same underlying graph but have different orientations isomorphic?
I know the fact that (undirected) exchange graphs of quivers with the same underlying undirected graph but have different orientations are isomorphic (i.e. quivers that are just finitely many arrow-...
- 156k
10
votes
1
answer
3k
views
Cluster algebras and cluster varieties
I have a really basic question about cluster algebras and cluster varieties. According to the definition of Fomin-Zelevinsky a cluster algebra is generated by a bunch of polynomial rings inside the ...
CommunityBot
- 1
1
vote
0
answers
99
views
Generalized Gaussian Decomposition
Let $G$ be a connected complex semisimple Lie group. Let $H$ be a maximal torus of $G$, let $W$ be the Weyl group of $G$, and let $N_\pm$ be a pair of opposite maximal unipotent subgroups. For each ...
- 540
6
votes
0
answers
340
views
cluster variables and L-functions
There is something in common between
cluster variables in the theory of cluster algebras,
L-functions in number theory,
namely the fact that both map direct sums to products, just like determinants ...
- 3,597
8
votes
0
answers
370
views
When is a $2$-Calabi–Yau triangulated category the cluster category of a QP?
Keller–Reiten's main theorem in Acyclic Calabi–Yau categories implies that if $\mathcal{C}$ is a $2$-Calabi–Yau (algebraic) triangulated category admitting a cluster-tilting object $T$ such that the ...
- 1,690
4
votes
0
answers
216
views
``Occasional'' Laurent phenomenon
This question is motivated by Richard Stanley's A question on the Laurent phenomenon (motivated by his answer to the question what is the probability that a scissor became the champion?).
He asked ...
CommunityBot
- 1
5
votes
2
answers
657
views
How to flip one triangulation on a surface into another
Let $S$ be a compact orientable surface and $p_1,\dots, p_n\in S$ be distinct points. We consider all triangulations on $S$ with vertices $p_1,\dots, p_n$.
Is there an algorithm which takes two ...
- 28.2k
8
votes
0
answers
224
views
Testing membership in a cluster algebra
Say I have a cluster algebra with principal coefficients and initial cluster $x_1,\ldots,x_n$. I don't want to invert the coefficient variables $y_1,\ldots,y_n$. The Laurent Phenomenon says that ...
- 1,846
3
votes
2
answers
362
views
Kahler differentials on cluster varieties
On affine toric varieties there is a classical theorem of Danilov which gives some combinatorial ways to describe the global sections of an appropriate sheaf of Kahler differentials as a vector space. ...
- 156k
1
vote
1
answer
617
views
Finding particular reduced words for Weyl group elements
I am studying cluster algebra structures on the coordinate rings of partial flag varieties, as defined in the paper Partial flag varieties and preprojective algebras by Geiss, Leclerc and Schröer. One ...
- 8,597
6
votes
0
answers
329
views
Non-crystallographic cluster algebras
Background
Fomin and Zelevinsky have introduced cluster algebras in an influential article. To define a cluster algebra, Fomin and Zelevinsky have defined a mutation of seeds. Here, a seed $(\mathbf{...
- 2,600
33
votes
2
answers
2k
views
What do cluster algebras tell us about Grassmannians?
One of the first examples of a cluster algebra given in Fomin and Zelevinsky's original paper is the homogeneous coordinate ring $\mathbb{C}[G_{2,n}]$ of the Grassmannian of planes in $\mathbb{C}^n$. ...
- 6,302
3
votes
0
answers
417
views
Motivic DT-Invariants for the Algebro-Geophobic
I am looking for as gentle of possible of an introduction to Kontsevich-Soibelman's theory of motivic DT-invariants. Specifically I am interested in the algebraic aspects of the theory and the ...
- 2,283