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Kontsevich and Soibelman defined the Donaldson-Thomas invariants of a 3dCalabi-Yau category with a stability condition. Any cluster variety can producean example of such a category, whose corresponding Donaldson-Thomas invariantsare encoded by a special formal automorphism of the cluster variety, known asthe Donaldson-Thomas transformation.  In this paper we prove a conjecture of Goncharov and Shen in the case of$\mathrm{GL}_n$, which describes the Donaldson-Thomas transformation of thedouble quotient of the double Bruhat cells $H \backslash \mathrm{GL}_n^{u,v}/H$where $H$ is a maximal torus, as a certain explicit cluster transformationrelated to Fomin-Zelevinsky's twist map. Our result, combined with the work ofGross, Hacking, Keel, and Kontsevich, proves the duality conjecture of Fock andGoncharov in the case of $H\backslash \mathrm{GL}_n^{u,v}/H$.

Donaldson-Thomas Transformation of Double Bruhat Cells in General Linear Groups