### 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 into **clusters**: subsets of cardinality $n$ (for some $n$). Each cluster is equipped with some combinatorial data which allows the complete set of cluster variables (and hence $A$) to be recovered from any given cluster. There is a related extension algebra $U$, called the **upper cluster algebra**, which coincides with the cluster algebra $A$ in small examples.

A fundamental example of this structure is a double Bruhat cell. Given a complex, reductive group $G$ with Borel $B$ and Weyl group $W$, there is a **Bruhat decomposition**
$$ G= \coprod_{w\in W} BwB$$
Choosing the opposite Borel $B^-$ instead gives the **opposite Bruhat decomposition**
$$ G=\coprod_{v\in W} B^-vB^-$$
and so
$$G = \coprod_{(w,v)\in W^2} BwB\cap B^-vB^-$$
The pieces $G_{w,v}$ of this disjoint union are called **double Bruhat cells**. Then Berenstein, Fomin and Zelevinsky have shown the following.

Theorem (BFZ05)The coordinate ring of a double Bruhat cell is naturally an upper cluster algebra.

### The Question

Roughly, my question is this. **If I have an upper cluster algebra, how can I tell if it is the coordinate ring of a double Bruhat cell?**

Note that there may be many different choices of $G,v,w$ which give isomorphic double Bruhat cells $G_{w,v}$, so it seems unlikely there is a constructive way of producing an example.

A more technical aspect of the question is the presence of **frozen variables**. These are cluster variables which cannot be mutated, and so they are in every cluster. Frozen variables may always be eliminated by setting them equal to 1, without affecting the combinatorics of the clusters.

The coordinate rings of double Bruhat cells will have many frozen variables, and I want to be able to eliminate or add frozen variables without changing the 'class' of cluster algebra. Though this is not a standard terminology, let us call two cluster algebras **equivalent** if they are related by the equivalence relation generated by 'setting some frozen variables' equal to 1.

Then I ask, **If I have an upper cluster algebra, how can I tell if it is equivalent to the coordinate ring of a double Bruhat cell?**

This second question is harder, but it does a better job of getting to the combinatorial type of the cluster algebra.