Tropical version of exchange relations in cluster algebras

Question

The exchange relation in a cluster algebra is \begin{align} x_k' = \frac{1}{x_k} (\prod_{j \to k}x_j + \prod_{k \to j} x_j). \end{align} Do we have some tropical version of this relation? Are there some references? Thank you very much.

Answer 1

This maybe too naive and not helpful to your problem: pretend $x_i$ are positive real number, and let $x_i = e^{-t a_i}$ for $a_i \in \mathbb{R}$. One can consider the limit $t \to \infty$ and taking $log$ on both sides to get $$ a_k' = \min(\sum_{j \to k} a_j, \sum_{k \to l} a_l) - a_k $$ Of course, this is the same thing as doing the tropical sum (the usual min) and tropical product/quotient (the usual $+/-$).

The paper by Gross-Hacking-Keel-Kontsevich, Canonical bases for cluster algebras, maybe is more relevant.

Answer 2

In the paper, the formula (2.4) gives a tropical version of mutation relations: \begin{align} a_k' = \max( \sum_i a_i[b_{ki}]_+, \sum_i a_i [-b_{ki}]_+ )-a_k. \end{align}