Semi-toric degenerations of Richardson varieties arising from cluster structures on flag varieties

A toric degeneration of an irreducible variety is a flat degeneration to an irreducible toric variety. In the case of a flag variety, its toric degeneration with desirable properties induces degenerations of Richardson varieties to unions of irreducible closed toric subvarieties, called semi-toric degenerations. For instance, Morier-Genoud proved that Caldero's toric degenerations arising from string polytopes have this property. Semi-toric degenerations are closely related to Schubert calculus. Indeed, Kogan-Miller constructed semi-toric degenerations of Schubert varieties from Knutson-Miller's semi-toric degenerations of matrix Schubert varieties which give a geometric proof of the pipe dream formula of Schubert polynomials. In this paper, we focus on a toric degeneration of a flag variety arising from a cluster structure, and prove that it induces semi-toric degenerations of Richardson varieties. Our semi-toric degeneration can be regarded as a generalization of Morier-Genoud's and Kogan-Miller's semi-toric degenerations.