Tropical totally positive cluster varieties

Abstract

We study the relation between the integer tropical points of a cluster variety (satisfying the full Fock-Goncharov conjecture) and the totally positive part of the tropicalization of an ideal presenting the corresponding cluster algebra. Suppose we are given a presentation of the cluster algebra by a Khovanskii basis for a collection of ${\bf g}$-vector valuations associated with several seeds related by mutations. In presence of a full rank fully extended exchange matrix we construct the rays of a subfan of the totally positive part of the tropicalization of the ideal that coincides combinatorially with the subgraph of the exchange graph of the cluster algebra corresponding to the collection of seeds. Moreover, geometric information about Gross-Hacking-Keel-Kontsevich's toric degenerations associated with seeds gets identified with the Gr\"obner toric degenerations obtained from maximal cones in the tropicalization. As application we prove a conjecture about the relation between Rietsch-Williams' valuations for Grassmannians arising from plabic graphs \cite{RW17} to Kaveh-Manon's work on valuations from the tropicalization of an ideal \cite{KM16}. In a second application we give a partial answer to the question if the Feigin-Fourier-Littelmann-Vinberg degeneration of the full flag variety in type $\mathtt A$ is isomorphic to a degeneration obtained from the cluster structure.