Seminar

A real matrix is totally positive if all of its minors are greater than zero, and, more generally, is totally nonnegative if all of its minors are nonnegative.

There is a cell decomposition of the space of totally nonnegative matrices of a given size, and the set of totally positive matrices forms the big cell in this decomposition. There are well-known efficient criteria for recognising total positivity, but until recently not for the other cells.

In this talk, I'll introduce some of the main ideas concerning totally nonnegative matrices, and then discuss the cell recognition problem, and give a solution that was motivated by connections with the theory of torus invariant prime ideals in quantum matrices.

This is joint work with Ken Goodearl and Stephane Launois.