Problem of the Week
Hilton Memorial Lecture
Well, actually, Alex Postnikov said “positroid” – don't blame me. These are matroids on [n] which can be realized over R by a matrix with all maximal minors nonnegative. They arise in the study of the totally nonnegative part of the Grassmannian, which in turn arises in applications in physics.
Closely related are positively oriented matroids. A positively oriented matroid is an oriented matroid of rank d on [n] which, up to reorientation, has a chirotope which is nonnegative on all increasing d-tuples. Thus a positroid could also be defined as the underlying matroid of a realizable positively oriented matroid.
I'll present a recent result of Ardila, Rincón, and Williams: all positively oriented matroids are realizable. Together with earlier work of da Silva, this leads to a simple combinatorial characterization of positroids. The proof makes elegant use of matroid basis polytopes.