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Plücker inequalities for weakly separated coordinates in totally nonnegative Grassmannian

Daniel Soskin, Prateek Kumar Vishwakarma

Abstract

We show that the partial sums of the long Plücker relations for pairs of weakly separated Plücker 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 Plücker 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 Plücker relations, and provides a very general and natural class of additive determinantal inequalities on the totally nonnegative part of the Grassmannian.

Plücker inequalities for weakly separated coordinates in totally nonnegative Grassmannian

Abstract

We show that the partial sums of the long Plücker relations for pairs of weakly separated Plücker coordinates oscillate around 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 Plücker 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 Plücker relations, and provides a very general and natural class of additive determinantal inequalities on the totally nonnegative part of the Grassmannian.
Paper Structure (9 sections, 10 theorems, 54 equations, 5 figures)

This paper contains 9 sections, 10 theorems, 54 equations, 5 figures.

Table of Contents

  1. Introduction and main results
  2. Prerequisites
  3. TNN Grassmannian and TNN matrices
  4. Immanants and nonnegativity
  5. Immanants for general product of two minors
  6. Generalized Laplace inequalities
  7. Generalization to pairs of weakly separated Plücker coordinates
  8. Inequalities for products of any two weakly separated Plücker coordinates
  9. Proof of reverse implication in Theorem \ref{['th:ws1']}

Key Result

Theorem 1.1

Let $1\leq d < n$ be integers. Suppose $P_d:=[1,d],$ and $Q_{dk} := [n-d,n]\setminus\{n-d+k\},$ for all $k\in [0,d].$ Then for all $n\times n$ totally nonnegative matrices $A.$

Figures (5)

  • Figure 1: Kauffman diagrams $K_1,$$K_1,$ and $K_2,$ respectively, as refered to in Example \ref{['ex:1']}.
  • Figure 2: Pre-matching diagrams for Examples \ref{['Ex:prematch1']} and \ref{['Ex:prematch2']} respectively.
  • Figure 3: Kauffman diagrams $K_1',$$K_1',$ and $K_2',$ respectively, as referred to in Example \ref{['ex:3']}.
  • Figure 4: Unique bi-colored matching in the trivial case: see Theorem \ref{['th:refined']}. On the extreme right, we have a rotated trivial diagram.
  • Figure 5: Describing intermediate stages in the proof of Theorem \ref{['th:refined']}

Theorems & Definitions (29)

  • Theorem 1.1: Fallat--Vishwakarma fallat2023inequalities
  • Definition 1.1
  • Theorem A
  • Remark 1.2: "Uniform" reformulation
  • Example 1.3
  • Theorem 2.2
  • proof
  • Theorem 2.1: Rhoades--Skandera RSkanTLImmp
  • Theorem 2.2: Rhoades--Skandera RSkanTLImmp
  • Example 2.3
  • ...and 19 more