Sign variation, the Grassmannian, and total positivity

JOURNAL OF COMBINATORIAL THEORY SERIES A

卷:145

Article

Karp, Steven N.¹ 

[1] Univ Calif Berkeley, Dept Math, Berkeley, CA 94720 USA

关键词: Sign variation;    Total positivity;    Totally nonnegative Grassmannian;    Amplituhedron;    Grassmann polytope;    Positroid;    Oriented matroid;   

DOI  :  10.1016/j.jcta.2016.08.003

来源: Elsevier

【 摘 要 】

The totally nonnegative Grassmannian is the set of k-dimensional subspaces V of R-n whose nonzero Plucker 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 Plucker coordinates), then the vectors in V change sign at most m times iff certain sequences of Plucker 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. (C) 2016 Elsevier Inc. All rights reserved.

【 授权许可】

Free