On subdivisions of the permutahedron and flags of lattice path matroids

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.