Sign variation, the Grassmannian, and total positivity

TL;DR

This paper extends the sign variation characterization from the totally nonnegative Grassmannian to the entire Grassmannian, providing new criteria, algorithms, and applications related to sign patterns and total positivity.

Contribution

It generalizes sign variation results to all Grassmannians, introduces an algorithm for perturbing non-generic subspaces, and applies findings to amplituhedron and positroid cell characterization.

Findings

Sign variation bounds characterize generic subspaces.

An algorithm perturbs non-generic to generic subspaces while preserving sign variation.

Results apply to oriented matroids and amplituhedron constructions.

Abstract

The totally nonnegative Grassmannian is the set of k-dimensional subspaces V of R^n whose nonzero Pluecker 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 Pluecker coordinates), then the vectors in V change sign at most m times iff certain sequences of Pluecker 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.