Arrangements and Amounts of Equal Minors in Totally Positive Matrices

Understanding the positive Grassmannian has been a highly significant question in mathematics for several decades. We discuss arrangements of equal minors in the totally positive Grassmannian. It was previously shown that arrangements of equals minors of largest value correspond to the simplices in Sturmfels triangulation. Here we discuss arrangements of equals minors of second largest value, and show that they correspond to the facets of Sturmfels triangulation. We then define the notion of cubical distance and obtain a general conjecture that provides a correspondence between minors of m-largest value and maximal simplices of certain cubical distance in Sturmfels triangulation. We prove this conjecture for the case m = 3, and also for certain cases in Gr(2, n).