# Questions tagged [cluster-algebras]

Questions related to cluster algebras, a class of commutative rings introduced around 2000 by Fomin and Zelevinsky, and nearby topics.

87
questions

**4**

votes

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### Positivity conjecture for Somos sequences

Let $\{s_n\}$ be the Somos-$4$ sequence, which is defined by $$s_{n+4}s_n=\alpha s_{n+3}s_{n+1}+\beta s_n^2.$$ It is known that $s_n$ is a Laurent polynomial: $s_n\in\mathbb{Z}[s_1^{\pm1}, \ldots, s_4^...

**12**

votes

**2**answers

188 views

### Quiver representations of type $D_n$ mutation class

I was wondering if there is a classification of the indecomposable quiver representations of (not necessarily acyclic) quivers that are mutation equivalent to the $D_n$ Dynkin diagram. Such quivers ...

**6**

votes

**1**answer

113 views

### Grassmannian cluster algebra of infinite type has no trees in its mutation class

The question is why the statement in the title is true (is it?).
To elaborate, recall that Grassmannian cluster algebra, according to Scott`s paper Grassmannians and Cluster Algebras, is the cluster ...

**1**

vote

**0**answers

57 views

### Defining cluster algebras of finite type $\mathrm{A}$ by generators and relations

Consider a cluster algebra of finite type $\mathrm{A}$. The set of all (distinct) cluster variables is of finite cardinality, denote it by $k$, for such algebra. Is it true that, for an arbitrary ...

**3**

votes

**0**answers

65 views

### Geometric description of a type $A$ cluster algebra with universal coefficients

In Thm. 12.4 in Fomin and Zelevinsky - Cluster algebras IV: Coefficients we are given a recipe for constructing a cluster algebra with universal coefficients. The recipe is given in terms of (almost ...

**2**

votes

**0**answers

61 views

### Berenstein-Fomin-Zelevinsky's Ininital seeds and initial seeds from Postnikov diagrams

In Cluster algebra III by Berenstein-Fomin-Zelevinsky, Theorem 2.10, for any pair of reduced words $(u,v)$, they constructed an initial seed for the cluster algebra $\mathbb{C}[B^{u,v}]$, where $B^{u,...

**5**

votes

**1**answer

153 views

### Is this Laurent phenomenon explained by invariance/periodicity?

In Chapter 4 of his Tracking the Automatic
Ant, David Gale
discusses three families of recursively defined sequences of numbers, all
due to Dana Scott and inspired by the Somos sequences:
Sequence 1. ...

**3**

votes

**0**answers

52 views

### Why does this cluster tilting object form a local slice?

I am reading the paper "Cluster automrphisms", here is the link: http://prospero.dmat.usherbrooke.ca/ibrahim/publications/Cluster_Automorphisms.pdf
In the proof of lemma 3.1 I am stuck: For ...

**2**

votes

**0**answers

46 views

### Multiplication formula in Grassmannian cluster categories

Grassmannian cluster categories are studied in A categorification of Grassmannian cluster algebras and Cluster categories from Grassmannians and root combinatorics. The category $CM(B_{k,n})$ of Cohen-...

**15**

votes

**4**answers

1k views

### Some interesting and elementary topics with connections to the representation theory?

I'm going to give a talk to talented high school seniors (for nearly 1.25-1.75 hours, maybe a little bit longer). They know some abstract algebra (groups, rings, modules...), linear algebra (...

**2**

votes

**1**answer

127 views

### Choice of a ground ring for cluster algebras

In order to define cluster algebra one needs to define its ground ring. In most cases, we take a group $P$ (often called a coefficient group) which is taken to be an abelian multiplicative group . ...

**1**

vote

**1**answer

138 views

### Cluster algebras and tropical points

Recently, I have been looking at some articles about bases for cluster algebras and came across the idea of tropical points. I should highlight here that unfortunately I have no background on ...

**2**

votes

**0**answers

54 views

### Piecewise linear $\sigma_i$ - notation question

In cluster algebra framework, in order to get root clusters, a modified version of a simple reflection is used. Define $\sigma_i:\Phi_{\geq -1} \to \Phi_{\geq -1}$ by setting:
$ \sigma_i(\alpha) =
\...

**18**

votes

**2**answers

851 views

### Is it possible that the GHKK canonical basis for cluster algebras is the Lusztig/Kashiwara dual canonical basis?

Gross-Hacking-Keel-Kontsevich (https://arxiv.org/abs/1411.1394) constructed a canonical basis (the so-called “theta basis”) for a cluster algebra, at least assuming it satisfies a certain ...

**11**

votes

**0**answers

276 views

### Scattering diagram for the cluster algebra $ \mathbb C [N]$

Five years ago, Gross-Hacking-Keel-Kontsevich made a major advance in the theory of cluster algebras, by constructing bases of cluster algebras in large generality. A key tool in their construction ...

**1**

vote

**1**answer

160 views

### Are there some relations between F-polynomials and theta functions?

F-polynomials are certain polynomials appears in the expansion formula of a cluster variable, see for example the formula (6.5) in cluster algebras IV. Theta functions in the paper correspond to ...

**3**

votes

**0**answers

92 views

### Does singularity confinement imply a fixed pattern of irreducible factors?

Consider a rational map
$f \colon (x_1,\ldots,x_n) \mapsto (P_1(x_1,\ldots,x_n),\ldots,P_n(x_1,\ldots,x_n))$, where the $P_i$ are rational functions. Via iteration this map defines a discrete ...

**13**

votes

**2**answers

539 views

### Integer but not Laurent sequences

Are there any sequence given by a recurrence relation:
$x_{n+t}=P(x_t,\cdots,x_{t+n-1})$, where $P$ is a positive Laurent Polynomial, satisfy:
if $x_0=\cdots=x_{n-1}=1$, then the sequence is only ...

**3**

votes

**1**answer

143 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

**0**answers

114 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 ...

**11**

votes

**0**answers

168 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

**0**answers

51 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,...

**27**

votes

**2**answers

943 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 ...

**6**

votes

**0**answers

72 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

**1**answer

181 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

**0**answers

154 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

**0**answers

117 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

**1**answer

119 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

**1**answer

252 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 ...

**13**

votes

**1**answer

951 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

**0**answers

55 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

**0**answers

120 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

**1**answer

101 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. ...

**8**

votes

**2**answers

544 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 ...

**3**

votes

**0**answers

192 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

**0**answers

94 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

**1**answer

95 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

**1**answer

511 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

**1**answer

420 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

**1**answer

148 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

**1**answer

127 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

**2**answers

210 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

**0**answers

111 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 = ...

**4**

votes

**2**answers

221 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 ...

**3**

votes

**1**answer

181 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

**2**answers

331 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

**2**answers

144 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 ...

**0**

votes

**1**answer

205 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

**1**answer

60 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

**1**answer

152 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))...