**38**

votes

**0**answers

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**

votes

**2**answers

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 $\mathbb{C}^n$. ...

**13**

votes

**0**answers

414 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 $2n$ ...

**12**

votes

**2**answers

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**

votes

**1**answer

435 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 ...

**8**

votes

**1**answer

434 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 ...

**8**

votes

**0**answers

243 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**

votes

**0**answers

95 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 ...

**6**

votes

**0**answers

107 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 ...

**6**

votes

**0**answers

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**

votes

**2**answers

405 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 ...

**5**

votes

**0**answers

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 ...

**4**

votes

**2**answers

334 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 ...

**4**

votes

**0**answers

269 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 ...

**3**

votes

**2**answers

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. ...

**3**

votes

**0**answers

110 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 ...

**3**

votes

**0**answers

288 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 ...

**1**

vote

**1**answer

170 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 ...

**1**

vote

**0**answers

122 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..)