Smooth torus quotients of Richardson varieties in the Grassmannian

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Let [Formula: see text] and [Formula: see text] be positive coprime integers with [Formula: see text]. Let [Formula: see text] denote the subgroup of diagonal matrices in [Formula: see text]. We study the GIT quotient of Richardson varieties [Formula: see text] in the Grassmannian Gr[Formula: see text] by [Formula: see text] with respect to a [Formula: see text]-linearized line bundle [Formula: see text] corresponding to the Plücker embedding. We give necessary and sufficient combinatorial conditions for the quotient variety [Formula: see text] to be smooth.