# Questions tagged [cluster-algebras]

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

79
questions

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### 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-...

**12**

votes

**3**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

107 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 . ...

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vote

**1**answer

93 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 ...

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**0**answers

53 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) =
\...

**16**

votes

**2**answers

674 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 ...

**10**

votes

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225 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

153 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 ...

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87 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

516 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

134 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 $x$?

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106 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 ...

**10**

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138 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

49 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,...

**24**

votes

**2**answers

820 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 ...

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vote

**1**answer

178 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 ...

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vote

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144 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{\...

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votes

**0**answers

95 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-...

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votes

**1**answer

112 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

239 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

772 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.)

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51 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 $\prod$ ...

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votes

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109 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....

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votes

**1**answer

92 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**

votes

**2**answers

428 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 ...

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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 ...

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votes

**0**answers

86 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

93 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

407 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?
...

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votes

**1**answer

377 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

141 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, ...

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votes

**1**answer

118 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

192 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

93 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 = ...

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votes

**2**answers

212 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 ...

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votes

**1**answer

151 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....

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votes

**2**answers

323 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

135 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

**1**answer

190 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 ...

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vote

**1**answer

58 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 \...

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votes

**1**answer

148 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

**1**answer

77 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

**1**answer

154 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.

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867 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}...

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**1**answer

107 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 ...

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votes

**0**answers

221 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 :
$$S_{12}S_{34}-S_{13}S_{24}+S_{14}S_{23}=0 .$$
...

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vote

**1**answer

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

**1**answer

202 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 & ...

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vote

**1**answer

239 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 $A_n$, $B_n$, $C_n$, $D_n$, $E_6, E_7, E_8$, $...