-coordinates in a Totally Nonnegative Grassmannian

Postnikov constructed a decomposition of a totally nonnegative Grassmannian (Grkn)≥0 into positroid cells. We provide combinatorial formulas that allow one to decide which cell a given point in (Grkn)≥ belongs to and to determine affine coordinates of the point within this cell. This simplifies Postnikov’s description of the inverse boundary measurement map and generalizes formulas for the top cell given by Speyer and Williams. In addition, we identify a particular subset of Plücker coordinates as a totally positive base for the set of non-vanishing Plücker coordinates for a given positroid cell. Postnikov [5] has described a cell decomposition of a totally nonnegative Grassmannian into positroid cells, which are indexed by Γ -diagrams; this decomposition is analogous to the matroid stratification of a real Grassmannian given by Gelfand, Goresky, MacPherson, and Serganova [2]. Postnikov also introduced a parametrization of each positroid cell using a collection of parameters which we call Γ -coordinates. In this paper, we give an explicit criterion for determining which positroid cell contains a given point in a totally nonnegative Grassmannian and explicit combinatorial formulas for the Γ -coordinates of a point. This generalizes the formulas of Speyer and Williams given for the top dimensional positroid cell [6], and provides a simpler description of Postnikov’s inverse boundary measurement map, which was given recursively in [5]. For a fixed positroid cell, our formulas are written in terms of a minimal set of Plücker coordinates, and this minimal set forms a totally positive base (in the sense of Fomin and Zelevinsky [1]) for the set of Plücker coordinates which do not vanish on the specified cell. 1. Positroid stratification and the boundary measurement map In this section, we review Postnikov’s positroid stratification of a totally nonnegative Grassmannian and his boundary measurement map. Let Grkn denote the Grassmannian of k-dimensional subspaces of R . A point x ∈ Grkn can be described by a collection of (projective) Plücker coordinates (PJ(x)), indexed by k-element subsets J ∈ ( [n] k ) . The totally nonnegative Grassmannian (Grkn)≥0 is the subset of points x ∈ Grkn such that all Plücker coordinates PJ(x) can be chosen to be nonnegative. In [2], the authors gave a decomposition of the Grassmannian Grkn into matroid strata. More precisely, for a matroid M ⊆ ( [n] k ) , let SM denote the subset of points Date: February 26, 2009.