Séminaire Lotharingien de Combinatoire, 95B.71 (2026), 12 pp.
Spencer Backman, Galen Dorpalen-Barry, Anastasia Nathanson, Ethan Partida and Noah Prime
Line Shellings of Bergman Fans
Abstract.
Given a matroid M and a building set B on its lattice of flats, we prove that the associated nested set complex N is shellable.
This generalizes a classical result of Björner that the order complex of the lattice of flats of a matroid is shellable (the case when B is the maximum building set), and strengthens a result of Feichtner-Müller that N is Cohen-Macaulay for arbitrary B.
Our approach is geometric in nature utilizing the Bergman fan ΣM,B, and is inspired by Bruggesser and Mani's line shellings of polytopes. We prove that, given a normal complex P for ΣM,B, as introduced by Ross and the third author, and a lexicographic weight vector ω, the order of the vertices of P induced by ω is a shelling order for N. As special cases, our result implies shellability of the polymatroid Bergman fan, and gives new proofs of shellability for the fine and augmented Bergman fans. We compare our shelling order with Björner's order for the order complex of the lattice of flats.
Received: November 15, 2025.
Accepted: February 15, 2026.
Final version: April 1, 2026.
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