Questions tagged [cluster-algebras]

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

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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-...
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3answers
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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 (...
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1answer
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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1answer
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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0answers
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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) = \...
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2answers
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 ...
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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 ...
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1answer
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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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 ...
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2answers
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 ...
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1answer
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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“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 ...
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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 ...
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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,...
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2answers
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 ...
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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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1answer
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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(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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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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1answer
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 ...
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1answer
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 ...
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1answer
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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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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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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1answer
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. ...
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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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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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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....
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1answer
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 ...
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1answer
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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1answer
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 ...
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1answer
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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1answer
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 ...
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2answers
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 ...
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0answers
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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2answers
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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1answer
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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2answers
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 ...
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2answers
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 ...
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1answer
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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1answer
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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1answer
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))...
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1answer
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 ...
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1answer
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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1answer
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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0answers
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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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 ...
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1answer
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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1answer
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 $A_n$, $B_n$, $C_n$, $D_n$, $E_6, E_7, E_8$, $...