  Double Bruhat cells $G^{u,v}$ were studied by Fomin and Zelevinsky. They
provide important examples of cluster algebras and cluster Poisson varieties.
Cluster varieties produce examples of 3d Calabi-Yau categories with stability
conditions, and their Donaldson-Thomas invariants, defined by Kontsevich and
Soibelman, are encoded by a formal automorphism on the cluster variety known as
the Donaldson-Thomas transformation. Goncharov and Shen conjectured in that for
any semisimple Lie group $G$, the Donaldson-Thomas transformation of the
cluster Poisson variety $H\backslash G^{u,v}/H$ is a slight modification of
Fomin and Zelevinsky's twist map. In this paper we prove this conjecture, using
crucially Fock and Goncharov's cluster ensembles and the amalgamation
construction. Our result, combined with the work of Gross, Hacking, Keel, and
Kontsevich, proves the duality conjecture of Fock and Goncharov in the case of
$H\backslash G^{u,v}/H$.
