Abstract
The totally nonnegative Grassmannian is the set of k-dimensional subspaces V of Rn whose nonzero Plücker coordinates all have the same sign. Gantmakher and Krein (1950) and Schoenberg and Whitney (1951) independently showed that V is totally nonnegative iff every vector in V, when viewed as a sequence of n numbers and ignoring any zeros, changes sign at most k−1 times. We generalize this result from the totally nonnegative Grassmannian to the entire Grassmannian, showing that if V is generic (i.e. has no zero Plücker coordinates), then the vectors in V change sign at most m times iff certain sequences of Plücker coordinates of V change sign at most m−k+1 times. We also give an algorithm which, given a non-generic V whose vectors change sign at most m times, perturbs V into a generic subspace whose vectors also change sign at most m times. We deduce that among all V whose vectors change sign at most m times, the generic subspaces are dense. These results generalize to oriented matroids. As an application of our results, we characterize when a generalized amplituhedron construction, in the sense of Arkani-Hamed and Trnka (2013), is well defined. We also give two ways of obtaining the positroid cell of each V in the totally nonnegative Grassmannian from the sign patterns of vectors in V.
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
Summary
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 (+, −, +)
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