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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29 votes
2 answers
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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 ...
Christian Gaetz's user avatar
23 votes
0 answers
1k 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}...
uvdose's user avatar
  • 593
11 votes
1 answer
973 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 fact,...
Hugh Thomas's user avatar
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55 votes
1 answer
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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 $\...
Gil Kalai's user avatar
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33 votes
2 answers
2k 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 $\mathbb{C}^n$. ...
Matthew Pressland's user avatar
13 votes
2 answers
575 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 ...
Sylvester W. Zhang's user avatar
7 votes
3 answers
632 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 $y$...
Steve's user avatar
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6 votes
0 answers
325 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 $(\mathbf{...
Philipp Lampe's user avatar
4 votes
0 answers
214 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 ...
Alexey Ustinov's user avatar
4 votes
1 answer
381 views

Cluster algebras of type A and X

I will base my question on Fock and Goncharov's paper Dual Teichmüller and lamination spaces. Let $S$ be a surface with boundaries, marked points on such boundaries, punctures and boundaries without ...
giulio bullsaver's user avatar
3 votes
1 answer
167 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))...
bing's user avatar
  • 331