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The main goal of this paper is to show that the (multi-homogeneous)coordinate ring of a partial flag variety $\mathbb{C} [G / P_K^{-}]$ admits acluster algebra structure if $G$ is any simply-connected semisimple complexalgebraic group. We use derivation properties and a special lifting map toprove that the cluster algebra structure $\mathcal{A}$ of the coordinate ring$\mathbb{C}[N_K]$ of a Schubert cell constructed by Goodearl and Yakimov can belifted, in an explicit way, to a cluster structure $\hat{\mathcal{A}}$ livingin the coordinate ring of the corresponding partial flag variety. Then we use aminimality condition to prove that the cluster algebra $\hat{\mathcal{A}}$ isindeed equal to $\mathbb{C}[G / P_K^{-}]$.

A cluster structure on the coordinate ring of partial flag varieties