On subdivisions of the permutahedron and flags of lattice path matroids

TL;DR

This paper explores how permutahedra can be subdivided into polytopes related to flags of lattice path matroids, identifying hyperplanes that produce these subdivisions and linking them to the nonnegative flag variety.

Contribution

It characterizes the coarsest hyperplane splits of permutahedra into flag matroid polytopes, specifically Bruhat Interval Polytopes, and describes the defining hyperplanes.

Findings

Coarsest subdivisions into LPFMs are hyperplane splits into Bruhat Interval Polytopes.

Hyperplanes intersecting permutahedron produce Bruhat Interval Polytopes.

Subdivisions correspond to points in the nonnegative flag variety.

Abstract

In this manuscript we study the subdivisions of the permutahedron $\Pi_n$ into two subpolytopes corresponding to flags of positroids, which are in particular flags of lattice path matroids (LPFMs). A subpolytope $P_{[u,v]}$ of $\Pi_n$ is a Bruhat Interval Polytope (BIP) if $P_{[u,v]}$ is the convex hull of all the permutations (viewed as points in $\RR^n$) in the interval $[u,v]$ in the Bruhat order of $\S_n$. We show that the coarsest subdivisions we obtain into LPFMs are the only subdivisions of $\Pi_n$ via hyperplane splits, into subpolytopes corresponding to BIPs. More specifically, we describe the hyperplanes whose intersection with $\Pi_n$ give rise to BIPs. Hence, these subdivisions are polytopes coming from points in the complete nonnegative flag variety.