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

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0answers
23 views

Identify a curve from bunch of numerical data [on hold]

I am trying to identify/compare a similar curve with the data I have. Data format: X, Y: (1, 0.01), (2, 0.02), (3, 0.03), (2, 0.04), (n, k), Lets say I have a plot or values which will generate a ...
6
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0answers
104 views

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 ...
3
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0answers
109 views

``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 ...
4
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2answers
328 views

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 ...
8
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0answers
229 views

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 ...
7
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0answers
94 views

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 ...
3
votes
2answers
257 views

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. ...
1
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1answer
164 views

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 ...
6
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0answers
140 views

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 ...
5
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0answers
147 views

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 ...
1
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0answers
121 views

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..)
38
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1k views

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 ...
29
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2answers
1k views

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 $\mathbb{C}^n$. ...
3
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0answers
286 views

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 ...
4
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0answers
268 views

“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 ...
13
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0answers
411 views

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 $2n$ ...
5
votes
2answers
404 views

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 ...
8
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1answer
426 views

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 ...
12
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2answers
2k views

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 ...
9
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1answer
432 views

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 ...