UBC Discrete Math Seminar: Jonathan Boretsky

The totally positive flag variety of rank r, defined by Lusztig, can be described as the set of rank r flags of real linear subspaces which can be represented by a matrix whose minors are all positive. We show that, for flag varieties of consecutive rank, this equals the subset of the flag variety with positive Plücker coordinates, yielding a straightforward condition to determine whether a flag is totally positive. This generalizes the well-established fact, proven independently by many authors including Rietsch, Talaska and Williams, Lam, and Lusztig, that the totally positive Grassmannian equals the subset of the Grassmannian with positive Plücker coordinates. While this result is algebro-geometric in nature, we will also spend time discussing the proof method, which is almost entirely combinatorial, relying heavily on graph theory and the combinatorics of the Bruhat order. We will discuss related results, proven using similar combinatorics, giving total positivity conditions for flag varieties of types B, C, and D. This talk is primarily based on joint work with Chris Eur and Lauren Williams.