Choice of a ground ring for cluster algebras

Question

In order to define cluster algebra one needs to define its ground ring. In most cases, we take a group $P$ (often called a coefficient group) which is taken to be an abelian multiplicative group . Sometimes $P$ is endowed with some additional binary operation of (auxiliary) addition $\oplus$ which is commutative, associative and distributive w.r.t to the multiplication in $P$ and turns $P$ into a semifield. Then the group ring $\mathbb{Z}P$ is used as a ground ring for a cluster algebra $\mathcal{A}$. My question is, why this particular choice of $P$? Is it because we then know that $P$ is torsion-free and so the group ring $\mathbb{Z}P$ is an integral domain and so then we can consider its field of fractions?

Answer 1

I will expand a bit in an answer. In the comments the asker expressed interested in all aspects on ground rings for cluster algebras. I have though some about this, but mostly it seems that $\mathbb{Z}P$ is taken as ground ground in practice.

1. I think in Cluster algebras I: Foundations one goal was to be as general as possible (Sergey Fomin has expressed this to me in personal communication). At this point the theory was brand new and the authors did not know what generality would be needed. Here is one of few places a ground ring $\mathbb{A} \subseteq \mathbb{Z}P$ is taken and a cluster algebra $\mathcal{A}_{\mathbb{A}}$ is looked at. It most places after $\mathbb{A} = \mathbb{Z}P$ is taken implicitly without any mentioned of $\mathbb{A}$. (Another place "extra" generality occurs in when sign-skew-symmetric exchagne matrices are used as opposed to just skew-symmetrizable matrices. It is still open to find other conditions which guarantee the sign-skew-symmetric property after mutating).
2. I don't really know of a use of a semifield other than (a) the tropical semifield that shows up in geometric type or (b) the universal semifield of subtraction few rational expressions. But having the semifield generality allows both these choices to be treated at the same type. I would be interested to see an application of some other semifield, I do not know of any such instances.
3. Taking ground ring $\mathbb{A} = \mathbb{Z}P$ makes some stuff work nicely, For example, many parts Muller's theory of Locally acyclic cluster algebras depends on frozen variables being invertible. With Bucher and Shapiro we investigate some of what happens if frozen variables are not inverted in Upper cluster algebras and choice of ground ring (I apologize for the self promotion, but I am excited someone is interested in ground ring!). In particular, we give an example where equality with the upper cluster algebra is sensitive to ground ring choice. Our motivation was that Gekhtman-Shapiro-Vainshtein have a cluster structure on $3 \times 3$ matrices with frozen variables not inverted (because frozen variables correspond to regular functions) in Cremmer--Gervais cluster structure on $SL_n$. An example I like to give for this idea is in cluster algebra structure on the Grassmannian the frozen vertices correspond to Plücker coordinates $\Delta_{i,i+1, \dots, i+k-1}$. Whether or not they are inverted changes between the Grassmannian and an open postroid variety. Goodearl and Yakimov also have a very nice paper titled Cluster algebra structures on Poisson nilpotent algebras Cluster algebra structures on Poisson nilpotent algebras which pays closed attention to if frozen variables are inverted or not.