Recently I begin working on matroids, in particular to a generalization of oriented matroids to the complex case. I found on arxiv the following interesting articles:

1)Alexander Postnikov: Total positivity, Grassmanians and networks

2)Federico Ardila, Felipe Rincón, Lauren Williams: Positively oriented matroids are realizable

3)Federico Ardila, Felipe Rincón, Lauren Williams: Positroids and non-crossing partitions http://arxiv.org/abs/1308.2698

I recall that a positroid of rank $k$ on $n$ elements is a matroid $M$ (of rank $k$ on $n$ elements) if there exist a real totally nonnegative $k\times n$ matrix which realize $M.$

In particular in (3) it is shown in detail a one-to-one correspondence between positroids of rank $k$ over $n$ elements and the cell decomposition of the totally nonnegative grassmanian $\mathcal{G}_{\mathbb{R}}^{\geq}(k,n).$ I hope not to make mistake with notations.

By the way, I'm interested in a generalization to the complex case. However, I have some problem in pointing out work, in particular I don't well understand what is the counterpart of the totally nonnegative Grassmanian in the complex case.