Abstract

The <i>totally nonnegative Grassmannian</i> is the set of $k$-dimensional subspaces $V$ of ℝ<sup>$n$</sup> whose nonzero Plücker coordinates (i.e. $k × k$ minors of a $k × n$ matrix whose rows span $V$) all have the same sign. Total positivity has been much studied in the past two decades from an algebraic, combinatorial, and topological perspective, but first arose in the theory of oscillations in analysis. It was in the latter context that Gantmakher and Krein (1950) and Schoenberg and Whitney (1951) independently showed that a subspace $V$ is totally nonnegative iff every vector in $V$, when viewed as a sequence of $n$ numbers and ignoring any zeros, changes sign fewer than $k$ times. We generalize this result, showing that the vectors in $V$ change sign fewer than $l$ times iff certain sequences of the Plücker coordinates of some <i>generic perturbation</i> of $V$ change sign fewer than $l − k + 1$ times. We give an algorithm which constructs such a generic perturbation. Also, we determine the <i>positroid cell</i> of each totally nonnegative $V$ from sign patterns of vectors in $V$. These results generalize to oriented matroids. La <i>grassmannienne totalement non négative</i> est l’ensemble des sous-espaces $V$ de ℝ<sup>$n$</sup> de dimension $k$ dont coordonnées plückeriennes non nulles (mineurs de l’ordre $k$ d’une matrice $k × n$ dont les lignes engendrent $V$) ont toutes le même signe. La positivité totale a beaucoup été étudiée durant les deux dernières décennies d’une perspective algébrique, combinatoire, et topologique, mais a pris naissance dans la théorie analytique des oscillations. C’est dans ce contexte que Gantmakher et Krein (1950) et Schoenberg et Whitney (1951) ont indépendamment démontré qu’un sous-espace $V$ est totalement non négatif ssi chaque vecteur dans $V$, lorsque considéré comme une séquence de $n$ nombres et dont on ignore les zéros, change de signe moins de $k$ fois. Nous généralisons ce résultat, démontrant que les vecteurs dans $V$ changent de signe moins de $l$ fois ssi certaines séquences des coordonnées plückeriennes d’une <i>perturbation générique</i> de $V$ changent de signe moins de $l − k + 1$ fois. Un algorithme construisant une telle perturbation générique est obtenu. De plus, nous déterminons la <i>cellule positroïde</i> de chaque $V$ totalement non négatif à partir des données de signe des vecteurs dans $V$. Ces résultats sont valides pour les matroïdes orientés.

Highlights

  • The Grassmannian Grk,n is the set of k-dimensional subspaces of Rn

  • In the 1930’s in oscillation theory in analysis, where positivity conditions on matrices can imply special oscillation and spectral properties. It was in this context that Gantmakher and Krein [GK50], and independently Schoenberg and Whitney [SW51], proved the first result about the totally nonnegative Grassmannian

  • The first generalizes Theorem 1.1 to all V ∈ Grk,n by giving a criterion for when the vectors in V change sign fewer than l times, in terms of the ∆I (V ). (Theorem 1.1 is the case l = k; see Corollary 4.7 for details.) The second shows, in the case that V is totally nonnegative, how to determine the cell of V in the cell decomposition of the totally nonnegative Grassmannian

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Summary

Introduction

The (real) Grassmannian Grk,n is the set of k-dimensional subspaces of Rn. Given V ∈ Grk,n, take a k × n matrix A whose rows span V ; for k-subsets I ⊆ {1, · · ·, n}, we let ∆I (V ) be the k × k minor. The first generalizes Theorem 1.1 to all V ∈ Grk,n by giving a criterion for when the vectors in V change sign fewer than l times, in terms of the ∆I (V ). The vectors in V ∈ Grk,n change sign fewer than l times iff there exists a generic perturbation V of V such that (∆J∪{i}(V ))i∈/J changes sign fewer than l − k + 1 times for all (k − 1)-subsets J ⊆ {1, · · ·, n}. By a certain algorithm (see Theorem 2.7) we can construct a generic perturbation W of V ∈ Grk,n such that the vectors in V change sign fewer than l times iff [1] holds for V = W. On which V realizes the alternating sign vector (+, −, +)

Sign changes and the Grassmannian
Positroids from sign patterns
1: A homeomorphism from
Results generalized to oriented matroids
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