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We show that the partial sums of the long Pl\"ucker relations for pairs ofweakly separated Pl\"ucker coordinates oscillate around $0$ on the totallynonnegative part of the Grassmannian. Our result generalizes the classicaloscillating inequalities by Gantmacher--Krein (1941) and recent results ontotally nonnegative matrix inequalities by Fallat--Vishwakarma (2023). In factwe obtain a characterization of weak separability, by showing that no otherpair of Pl\"ucker coordinates satisfies this property.  Weakly separated sets were initially introduced by Leclerc and Zelevinsky andare closely connected with the cluster algebra of the Grassmannian. Moreover,our work connects several fundamental objects such as weak separability,Temperley--Lieb immanants, and Pl\"ucker relations, and provides a very generaland natural class of additive determinantal inequalities on the totallynonnegative part of the Grassmannian.

Pl\"ucker inequalities for weakly separated coordinates in totally nonnegative Grassmannian