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

We show that the partial sums of the long Pl\"ucker relations for pairs of weakly separated Pl\"ucker coordinates oscillate around $0$ on the totally nonnegative part of the Grassmannian. Our result generalizes the classical oscillating inequalities by Gantmacher--Krein (1941) and recent results on totally nonnegative matrix inequalities by Fallat--Vishwakarma (2023). In fact we obtain a characterization of weak separability, by showing that no other pair of Pl\"ucker coordinates satisfies this property. Weakly separated sets were initially introduced by Leclerc and Zelevinsky and are 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 general and natural class of additive determinantal inequalities on the totally nonnegative part of the Grassmannian.