Polyhedral and tropical geometry of flag positroids

A flag positroid of ranks r := (r 1 < • • • < r k ) on n is a flag matroid that can be realized by a real r k × n matrix A such that the r i × r i minors of A involving rows 1, 2, . . ., r i are nonnegative for all 1 ≤ i ≤ k.In this paper we explore the polyhedral and tropical geometry of flag positroids, particularly when r := (a, a + 1, . . ., b) is a sequence of consecutive numbers.In this case we show that the nonnegative tropical flag variety TrFl ≥0 r,n equals the nonnegative flag Dressian FlDr ≥0 r,n , and that the points µ = (µa, . . ., µ b ) of TrFl ≥0 r,n = FlDr ≥0 r,n give rise to coherent subdivisions of the flag positroid polytope P (µ) into flag positroid polytopes.Our results have applications to Bruhat interval polytopes: for example, we show that a complete flag matroid polytope is a Bruhat interval polytope if and only if its (≤ 2)-dimensional faces are Bruhat interval polytopes.Our results also have applications to realizability questions.We define a positively oriented flag matroid to be a sequence of positively oriented matroids (χ 1 , . . ., χ k ) which is also an oriented flag matroid.We then prove that every positively oriented flag matroid of ranks r = (a, a + 1, . . ., b) is realizable.