A real matrix is totally nonnegative if each of its square submatrices has nonnegative determinant, and is totally positive if these determinants are all positive. While these definitions are easily understood, the theory of totally nonnegative matrices is rich and has influenced many other areas such as probability, networks and representation theory. Most recently, combinatorial aspects of total positivity have played a central role in the study of scattering amplitudes in planar four-dimensional theories and in the study of soliton solutions of the KP equation (a two-dimensional nonlinear dispersive wave equation). Thus, the theory of totally nonnegative matrices arose in a broad range of applications and has also led Fomin and Zelevinsky to the definition of cluster algebras. The importance of total positivity is emphasized by the recent publication of two books on the subject.
Our interest in this area arises because of an intriguing connection between total positivity and quantum groups. With collaborators we have been investigating this connection in the past few years. This new approach led, for instance, to new efficient criteria for a real matrix to be totally nonnegative. The aim of this Research in Pairs event is two-fold. First, we will
continue developing our approach to total positivity through noncommutative algebra tools, and vice-versa. Next, our second aim will be to make progress towards producing a monograph encompassing much of the recent developments in the study of totally nonnegative matrices.

Participants

Stéphane Launois (University of Kent)
Tom Lenagan (University of Edinburgh)

Sponsor