Papers › Polyhedral and Tropical Geometry of Flag Positroids
Polyhedral and Tropical Geometry of Flag Positroids
Jonathan Boretsky, Christopher Eur, Lauren Williams
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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 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_(r,n)^(≥0) equals the nonnegative flag Dressian FlDr_(r,n)^(≥0), and that the points μ = (μₐ,…, μ_b) of TrFl_(r,n)^(≥0) = FlDr_(r,n)^(≥0) 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 (χ₁,…,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.
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