## 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 $\{x_1,...,x_n\}$ by a certain recursion determined by a skew-symmetrizable matrix $M$ (**mutation**).

The mutation of the pair (called a *seed*)
$$((x_1,x_2,...,x_i,...,x_n),M)$$
at the $i$-th place is the seed
$$((x_1,x_2,...,\mu_i(x_i),...,x_n),\mu_i(M))$$

$$\mu_i(x_i)=\left[\prod_{j,M_{ij}>0} x_j^{M_{ij}}+\prod_{j,M_{ij}<0}x_j^{-M_{ij}}\right]x_i^{-1}$$ and $\mu_i(M)$ is a new skew-symmetrizable matrix I cannot get mathjax to display.

Mutation of seeds may be iterated arbitrarily. It is a non-trivial theorem that the functions in the seed are always Laurent polynomials. A cluster variable is any function appearing in a seed obtained by a sequence of mutations.

## Ring-theoretic properties of cluster variables

As elements in $A$, each cluster variable $f$ is an irreducible element (it can't be factored). This is not hard to show; it follows from the Laurent embedding for any cluster containing $f$, and the observation that $A$ does not contain any Laurent monomials with negative powers of (non-frozen) variables.

So then I ask, **are cluster variables prime elements in $A$?**

The analogous argument to the irreducible case doesn't seem to work, since one needs to consider more general Laurent polynomials in a given cluster, and there is no nice criterion for telling when a general Laurent polynomial is in the cluster algebra. It also seems unlikely that $A$ is a UFD in general, which would be a standard trick for deducing primality for irreducibility. Nonetheless, the examples I checked in Macaulay 2 were all prime.