Questions tagged [cluster-algebras]
Questions related to cluster algebras, a class of commutative rings introduced around 2000 by Fomin and Zelevinsky, and nearby topics.
3
votes
1answer
107 views
“Solution” of finite cluster algebras
Consider the cluster algebras $A_n$ and $D_n$. Choose any cluster $x$, is there an explicit formula that express all other cluster variables in terms of $x$?
8
votes
0answers
78 views
“Cross-Ratios” for D_n cluster algebra
Both cluster algebras $A_n$ and $D_n$ admit an interpretation in terms of (tagged) triangulations of Riemann surfaces - respectively a Poincaré disk with n+3 punctures on the boundary and a Poincaré ...
9
votes
0answers
103 views
Cluster algebra and Fenchel Nielsen coordinates
Certain cluster algebras arise from ideal triangulations of hyperbolic Riemann surfaces. The combinatorics behind their mutations can be understood in terms of "flips" in the triangulation, and the ...
2
votes
0answers
48 views
Differential (in)dependancies between cluster variables
Let $ \mathcal A$ be the complex cluster algebra
obtained from the initial seed $(a,x, B)$ where :
$x$ stands for a $n$-tuple $(x_1,\ldots,x_n)$ of complex indeterminates, idem for $a=(a_1,\ldots,...
23
votes
2answers
735 views
Determining if a rational function has a subtraction-free expression
This question was first asked by Mehtaab Sawhney in Alex Postnikov's combinatorics class.
Given a rational function $F=P(x_1,...,x_n)/Q(x_1,...,x_n)$ with (say) integer coefficients, it is often of ...
5
votes
0answers
66 views
Non-rigid indecomposable summands of simple-minded collections in bounded derived category of hereditary algebras
Let $\Lambda$ be a hereditary algebra over an algebraically closed field $k$. Let $S$ be one of the indecomposable summands of one simple-minded collection in $D^b(\Lambda)$. Is it true that $S$ is ...
1
vote
1answer
159 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 ...
1
vote
0answers
135 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{\...
0
votes
0answers
81 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-...
4
votes
1answer
95 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 ...
6
votes
1answer
212 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 ...
12
votes
1answer
425 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.)
2
votes
0answers
46 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 $\prod$ ...
2
votes
0answers
96 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....
6
votes
1answer
79 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. ...
6
votes
2answers
322 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 ...
1
vote
0answers
117 views
Definition of loop amplituhedrons
In the paper The Amplituhedron
, Nima Arkani-Hamed and Jaroslav Trnka introduced the geometric object amplituhedron. It is defined as follows (see also the lecture notes).
Let $Z$ be a $(k+m)\times ...
2
votes
0answers
75 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....
4
votes
1answer
81 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 ...
3
votes
1answer
252 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?
...
13
votes
1answer
309 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
votes
1answer
127 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, ...
3
votes
1answer
108 views
Reference request: coordinate ring of $OSP(2p|n)$
In the paper, the orthosymplectic supergroup $OSP(2p|n)$ is defined as follows.
Let $A = A_0 \oplus A_1$ be a supercommutative superalgebra, where elements in $A_0$ are even and elements in $A_1$ are ...
3
votes
2answers
159 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 ...
2
votes
0answers
67 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 = ...
3
votes
2answers
196 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 ...
2
votes
1answer
138 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....
4
votes
2answers
308 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 ...
0
votes
2answers
123 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 ...
-1
votes
1answer
175 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 ...
1
vote
1answer
56 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 \...
3
votes
1answer
142 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))...
1
vote
1answer
74 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 ...
0
votes
1answer
147 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.
20
votes
0answers
700 views
Is A276175 integer-only?
The terms of the sequence A276123, defined by $a_0=a_1=a_2=1$ and $$a_n=\dfrac{(a_{n-1}+1)(a_{n-2}+1)}{a_{n-3}}\;,$$ are all integers (it's easy to prove that for all $n\geq2$, $a_n=\frac{9-3(-1)^n}{2}...
1
vote
1answer
91 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 ...
4
votes
0answers
215 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 :
$$S_{12}S_{34}-S_{13}S_{24}+S_{14}S_{23}=0 .$$
...
1
vote
1answer
95 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 ...
0
votes
1answer
168 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 & ...
1
vote
1answer
205 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 $A_n$, $B_n$, $C_n$, $D_n$, $E_6, E_7, E_8$, $...
4
votes
1answer
145 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 ...
2
votes
1answer
166 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 ...
5
votes
1answer
246 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 ...
3
votes
1answer
195 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-...
2
votes
0answers
58 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 ...
3
votes
1answer
259 views
Do we have super Plucker relations for a super Grassmannian?
Super Grassmannians are introduced by Manin, see for example. We have Plucker relation for Grassmannian.
Are there some references about super Plucker relations for super Grassmannian? Thank you ...
1
vote
1answer
118 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-...
7
votes
1answer
1k 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 ...
1
vote
0answers
71 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 ...
4
votes
3answers
182 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 ...
