Line Shellings of Geometric Lattices
Spencer Backman, Galen Dorpalen-Barry, Anastasia Nathanson, Ethan Partida, Noah Prime
TL;DR
The paper addresses the shellability of the order complex and, more generally, of nested set complexes for matroids by transporting line-shelling ideas from polytopes to Bergman fans. It builds a bridge between tropical-geometric realizations (Bergman fans and normal complexes) and combinatorial orderings (NL-order) to produce a shelling via a lexicographic vertex-order on a cubical normal complex. The main contribution is a uniform line-shelling proof (Theorem 1) that applies to all nested set complexes arising from matroids, supplementing Björner’s EL-shelling framework and clarifying the relationship between combinatorial labeling and geometric shellings. This work deepens the connection between tropical geometry, building sets, and shellability, with potential implications for Cohen–Macaulay properties and the broader understanding of matroidal fans.
Abstract
Inspired by Bruggesser-Mani's line shellings of polytopes, we introduce line shellings for the lattice of flats of a matroid: given a normal complex for a Bergman fan of a matroid induced by a building set, we show that the lexicographic order of the coordinates of its vertices is a shelling order. This gives a new proof of Björner's classical result that the order complex of the lattice of flats of a matroid is shellable, and demonstrates shellability for all nested set complexes for matroids.