Sign variation, the Grassmannian, and total positivity

Steven N. Karp

Published 2015-03-19Version 1

The totally nonnegative Grassmannian is the set of $k$-dimensional subspaces $V$ of $\mathbb{R}^n$ whose nonzero Pl\"ucker 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 Pl\"ucker coordinates), then the vectors in $V$ change sign at most $m$ times iff certain sequences of Pl\"ucker 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. 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 $V$. These results generalize to oriented matroids.