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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 := (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,n0 equals the nonnegative flag Dressian FlDr r,n0, and that the points μ = (μa,,μb) of TrFl r,n0 = FlDr r,n0 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.

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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