Séminaire Lotharingien de Combinatoire, 89B.60 (2023), 12 pp.
Jonathan Boretsky, Christopher Eur and Lauren Williams
Polyhedral and Tropical Geometry of Flag Positroids
Abstract.
A flag positroid of ranks r := (r1<...<rk) on [n]
is a flag matroid that can be realized by a real rk × n matrix A
such that the ri × ri minors of A
involving rows 1,2,...,ri are nonnegative for all 1≤i ≤k. In this abstract, 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
TrFlr,n≥0 equals the
nonnegative flag Dressian FlDrr,n≥0, and that the points
μ = (mu;a,...,μb) of
TrFlr,n≥0 =
FlDrr,n≥0 give rise to coherent subdivisions of flag positroid polytopes into (smaller) 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 (&chi1,...,χk) 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.
Received: November 15, 2022.
Accepted: February 20, 2023.
Final version: April 1, 2023.
The following versions are available: