# 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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### 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. ...
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### 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 ...
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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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### 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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### 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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### 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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### 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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### 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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### 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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### 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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### 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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### 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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### 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 ...
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 $(k+m)\times ... 0answers 94 views ### Weighted quiver in Keller's Java Quiver mutation [closed] I need to do mutation using weighted quivers. There are weights on the arrows which looks like: How to draw weighted quiver in Keller's Java Quiver mutation: https://webusers.imj-prg.fr/~bernhard.... 1answer 95 views ### Rigid regular objects of path algebras of tame quivers In the paper On Maximal Green Sequences by Brustle, Dupont and Perotin the authors argued that in a path algebra$\Lambda=kQ$of a tame quiver$Q$with$n$vertices each tilting module contains at ... 1answer 511 views ### What are the relations among canonical basis, dual canonical basis, Semicanonical Basis, dual semicanonical bases? I am reading the lecture notes and would like to know more about canonical basis. What are the relations among canonical basis, dual canonical basis, Semicanonical Basis, dual semicanonical bases? ... 1answer 420 views ### Is there some relation between cluster algebras and crystal graphs? Cluster algebras are closely related to totally positivity in algebraic groups and canonical bases in quantum groups. Is there some relation between cluster algebras and crystal graphs? Can the ... 1answer 148 views ### Which cluster algebras where the existence of maximal green sequences is still unknown? Maximal green sequences are studied in many papers. For example, Maximal Green Sequences for Cluster Algebras Associated to the n-Torus by Eric Bucher, On Maximal Green Sequences by Thomas Brüstle, ... 1answer 127 views ### Reference request: coordinate ring of$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_1are ... 2answers 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 ... 0answers 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 = ... 2answers 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 ... 1answer 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.... 2answers 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 ... 2answers 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 ... 1answer 205 views ### What are the cluster algebra structures onGr(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)$)? Do we have ... 1answer 60 views ### A question about exchange pattern Exchange pattern, see Section 2 in "cluster algebras I: foundations" by Fomin and Zelevinsky or How to understand exchange pattern? Given an example$\cdots \overset{2}{-} t_1 \overset{1}{-} t_2 \...
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))...