Combinatorial formulas for -coordinates in a totally nonnegative Grassmannian

Postnikov constructed a decomposition of a totally nonnegative Grassmannian ( Gr k n ) ⩾ 0 into positroid cells. We provide combinatorial formulas that allow one to decide which cell a given point in ( Gr k n ) ⩾ 0 belongs to and to determine affine coordinates of the point within this cell. This simplifies Postnikovʹs 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.