Combinatorics of totally positive matrices

A matrix is called totally positive if all its minors are positive. How many minors of an $n$ by $n$ matrix must be tested so that their positivity would imply the total positivity of the matrix? How to construct such minimal total positivity criteria?
These questions are related to the problem of factoring an invertible square matrix into the minimal number of “elementary factors”. The solutions involve combinatorics of reduced words and pseudo-line arrangements, together with a family of biregular automorphisms of “double Bruhat cells” in the general linear group. This is joint work with Andrei Zelevinsky.