Totally nonnegative tropical flags and the totally nonnegative flag Dressian

We study the totally nonnegative part of the complete flag variety and of its tropicalization. We show that Lusztig’s notion of nonnegative complete flag variety coincides with the set of complete flags which have nonnegative Plücker coordinates. This mirrors the characterization of the totally nonnegative Grassmannian as the set of points in the Grassmannian with nonnegative Plücker. We then study the tropical complete flag variety and complete flag Dressian, which are two tropical versions of the complete flag variety, capturing realizable and abstract flags of tropical linear spaces, respectively. In general, the complete flag Dressian properly contains the tropical complete flag variety. However, we show that the totally nonnegative parts of these spaces coincide.