On subdivisions of the permutahedron and flags of lattice path matroids

TL;DR

The paper characterizes coarse subdivisions of the permutahedron $Π_n$ into Bruhat interval polytopes (BIPs) that arise from flags of lattice path flag matroids (LPFMs). It proves that the coarsest such subdivisions come from hyperplane splits parallel to facet-defining hyperplanes, yielding subpolytopes that are flag matroid polytopes of LPFMs and correspond to points in the complete nonnegative flag variety $Fl_n^{\geq 0}$. Moreover, these hyperplanes (types (T1)-(T3)) are exactly the hyperplanes that produce any split of $Π_n$ into BIPs, establishing a complete classification of coarse LPFM-induced splits and revealing a link to tropical geometry via $Tr^{>0}Fl_n$. The work provides a structured framework for constructing and understanding LPFM-based Bruhat interval subdivisions, and it poses several questions about the full lattice of subdivisions and their tropical interpretations.

Abstract

In this manuscript we study the subdivisions of the permutahedron $Π_n$ into two subpolytopes corresponding to flags of positroids, which are in particular flags of lattice path matroids (LPFMs). A subpolytope $P_{[u,v]}$ of $Π_n$ 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 $§_n$. We show that the coarsest subdivisions we obtain into LPFMs are the only subdivisions of $Π_n$ via hyperplane splits, into subpolytopes corresponding to BIPs. More specifically, we describe the hyperplanes whose intersection with $Π_n$ give rise to BIPs. Hence, these subdivisions are polytopes coming from points in the complete nonnegative flag variety.

Figures (9)

• Figure 1: A basis in the diagram of the LPM $M[1246,3568]$.
• Figure 2: Left: $M=M[1247,3568]$. Center: elementary quotient of $M$. Right: a (not elementary) quotient of $M$.
• Figure 3: Left: $\Pi_4$ obtained as the uniform flag. Right: an LPFM polytope with two of its vertices highlited.
• Figure 4: $\ell_1$ is a good pair with either $2, 6, 8$.
• Figure 5: Left: Bad $H$-split with $H:x_1+x_2=5$. Right: Bad $H$-split with $H:x_3=3$

Theorems & Definitions (35)

• Definition 3.1
• Definition 3.2
• Definition 3.3
• Remark 3.4
• Definition 3.5
• Example 3.6
• Remark 3.7
• Definition 3.8
• Remark 3.9
• Definition 3.10