Polyhedral and tropical geometry of flag positroids

Boretsky, Jonathan; Eur, Christopher; Williams, Lauren

doi:10.2140/ant.2024.18.1333

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Polyhedral and tropical geometry of flag positroids

Jonathan Boretsky, Christopher Eur and Lauren Williams

Vol. 18 (2024), No. 7, 1333–1374

DOI: 10.2140/ant.2024.18.1333

Abstract

A flag positroid of ranks $r : = (r_{1} < \dots < r_{k})$ on $[n]$ is a flag matroid that can be realized by a real $r_{k} \times n$ matrix $A$ such that the $r_{i} \times r_{i}$ minors of $A$ involving rows $1, 2, \dots, r_{i}$ are nonnegative for all $1 \leq i \leq k$ . In this paper we explore the polyhedral and tropical geometry of flag positroids, particularly when $r : = (a, a + 1, \dots, b)$ is a sequence of consecutive numbers. In this case we show that the nonnegative tropical flag variety ${TrFl}_{r, n}^{\geq 0}$ equals the nonnegative flag Dressian ${FlDr}_{r, n}^{\geq 0}$ , and that the points $μ = (μ_{a}, \dots, μ_{b})$ of ${TrFl}_{r, n}^{\geq 0} = {FlDr}_{r, n}^{\geq 0}$ give rise to coherent subdivisions of the flag positroid polytope $P (\underline{μ})$ 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 $(\leq 2)$ -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 $(χ_{1}, \dots, χ_{k})$ which is also an oriented flag matroid. We then prove that every positively oriented flag matroid of ranks $r = (a, a + 1, \dots, b)$ is realizable.

Keywords

flag variety, flag positroid, positroid polytope, Bruhat interval polytope, flag Dressian, positively oriented flag matroid, tropical flag variety

Mathematical Subject Classification

Primary: 05Exx

Milestones

Received: 16 September 2022

Revised: 10 July 2023

Accepted: 3 September 2023

Published: 13 June 2024

Authors

Jonathan Boretsky
Department of Mathematics Harvard University Cambridge, MA United States

Christopher Eur
Department of Mathematics Harvard University Cambridge, MA United States

Lauren Williams
Department of Mathematics Harvard University Cambridge, MA United States

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Vol. 18, No. 7, 2024