Séminaire Lotharingien de Combinatoire, 93B.86 (2025), 12 pp.
Daniel Soskin and Prateek Kumar Vishwakarma
Plücker Inequalities for Weakly Separated Coordinates in TNN Grassmannians
Abstract.
The fundamental connections of the Grassmannian with both weak separability and Plücker relations are well known. In this work we focus on the totally nonnegative (TNN) part of the Grassmannian, and discover the intrinsic connection between weak separability and Plücker relations. In particular, we show that certain natural sums of terms in a long Plücker relation for pairs of weakly separated Plücker coordinates oscillate around 0 over the TNN Grassmannian. This generalizes the classical oscillating inequalities by Gantmacher-Krein (1941) and recent results on TNN matrix inequalities by Fallat-Vishwakarma (2024). In fact we obtain a characterization of weak separability, by showing that no other pairs of Plücker coordinates satisfy this property. Moreover, our work uncovers the natural connections between weak separability, Plücker relations, as well as Temperley-Lieb immanants, and provides a general and natural class of additive inequalities in Plücker coordinates on the totally nonnegative part of the Grassmannian.
Received: November 15, 2024.
Accepted: February 15, 2025.
Final version: April 1, 2025.
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