Sign variation, the Grassmannian, and total positivity

The totally nonnegative Grassmannian is the set of k-dimensional subspaces V   of RnRn 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−1k−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+1m−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.