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

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Intersecting Family of Triangulations

Let $\cal T_n$ be the family of all triangulations on an $n$-gon using $(n-3)$ non-intersecting diagonals. The number of triangulations in $\cal T_n$ is $C_{n-2}$ the $(n-2)$th Catalan number. Let $\...
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What do cluster algebras tell us about Grassmannians?

One of the first examples of a cluster algebra given in Fomin and Zelevinsky's original paper is the homogeneous coordinate ring $\mathbb{C}[G_{2,n}]$ of the Grassmannian of planes in $\mathbb{C}^n$. ...
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Which cluster algebras have been categorified?

In "Tilting Theory and Cluster Combinatorics" Buan, Marsh, Reineke, Reiten, and Todorov constructed cluster categories for mutation finite cluster algebras (without coefficients), and Amiot gives a ...
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Catalan objects associated to a univariate polynomial

Given a monic degree $n$ polynomial $f(z)$ with no double roots, and a phase $0\leq \theta < \pi$, there are natural constructions which associate to this data: a noncrossing matching on $2n$ ...
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Which cluster algebras are coordinate rings of double Bruhat cells?

Background A uselessly vague paragraph follows. A cluster algebra is a commutative algebra $A$ with a distinguished set of generators called cluster variables. These cluster variables are grouped ...
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Are cluster variables prime elements?

Cluster algebras introduction A cluster algebra is a subalgebra $A$ of $k[x_1^{\pm1},...,x_n^{\pm1}]$ generated by a set of cluster variables, which are elements which can be generated from the set $\...
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What is a good introduction to cluster algebras from surfaces?

What is a good reference for cluster algebras from surfaces, with a view to their connection to Teichmuller theory? In my view, that means it should start off with unpunctured surfaces (and in fact,...
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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 ...
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Testing membership in a cluster algebra

Say I have a cluster algebra with principal coefficients and initial cluster $x_1,\ldots,x_n$. I don't want to invert the coefficient variables $y_1,\ldots,y_n$. The Laurent Phenomenon says that ...
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When is a $2$-Calabi–Yau triangulated category the cluster category of a QP?

Keller–Reiten's main theorem in Acyclic Calabi–Yau categories implies that if $\mathcal{C}$ is a $2$-Calabi–Yau (algebraic) triangulated category admitting a cluster-tilting object $T$ such that the ...
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cluster variables and L-functions

There is something in common between cluster variables in the theory of cluster algebras, L-functions in number theory, namely the fact that both map direct sums to products, just like ...
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Non-crystallographic cluster algebras

Background Fomin and Zelevinsky have introduced cluster algebras in an influential article. To define a cluster algebra, Fomin and Zelevinsky have defined a mutation of seeds. Here, a seed $(\mathbf{...
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Do Denominator Vectors Determine the Cluster Variable

Given a cluster algebra $A=A(\mathbf{x},Q)$, the Laurent Phenomenon states that all the cluster variables of $A$ are Laurent polynomials in the elements of $\mathbf{x}$. Thus, any cluster variable $y$...
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“Natural” Poisson structure on $(\mathbb{P}^1)^N$

Recently there is some interest in the Poisson geometry of "cluster manifolds", which are varieties associated to cluster algebras. See for example the works of Gekhtman, Shapiro and Vainshtein. In ...
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How to flip one triangulation on a surface into another

Let $S$ be a compact orientable surface and $p_1,\dots, p_n\in S$ be distinct points. We consider all triangulations on $S$ with vertices $p_1,\dots, p_n$. Is there an algorithm which takes two ...
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3answers
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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 ...
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``Occasional'' Laurent phenomenon

This question is motivated by Richard Stanley's A question on the Laurent phenomenon (motivated by his answer to the question what is the probability that a scissor became the champion?). He asked ...
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Kahler differentials on cluster varieties

On affine toric varieties there is a classical theorem of Danilov which gives some combinatorial ways to describe the global sections of an appropriate sheaf of Kahler differentials as a vector space. ...
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1answer
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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 ...
3
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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-...
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Motivic DT-Invariants for the Algebro-Geophobic

I am looking for as gentle of possible of an introduction to Kontsevich-Soibelman's theory of motivic DT-invariants. Specifically I am interested in the algebraic aspects of the theory and the ...
2
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1answer
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Number of cluster variables

In the paper cluster algebras and quantum affine algebras, Section 13.5, it is said that when $\mathfrak{g}$ is of type $A_2$ and $\ell=2$, then the corresponding cluster algebra $\mathscr{A}_2$ for $...
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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 ...
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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 ...
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1answer
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Finding particular reduced words for Weyl group elements

I am studying cluster algebra structures on the coordinate rings of partial flag varieties, as defined in the paper Partial flag varieties and preprojective algebras by Geiss, Leclerc and Schröer. One ...
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1answer
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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 ...
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1answer
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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-...
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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 ...
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Mutation is an involution

Let $Q$ be a finite quiver and let $\mu_{k}$ denote mutation at a vertex $k$. Why is this an involution? I don't see it clear (should be easy though..)
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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 ...