Georgia Institute of Technology College of Sciences

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. 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. We discuss the "tropicalization" of this result, relating the nonnegative tropical flag variety to the nonnegative Dressian, a space parameterizing the regular subdivisions of flag positroid polytopes into flag positroid polytopes. Many results can be generalized to flag varieties of types B and C. This talk is primarily based on joint work with Chris Eur and Lauren Williams and joint work with Grant Barkley, Chris Eur and Johnny Gao.