Decompositions of Augmented Bergman Complexes

TL;DR

This paper proves that augmented Bergman complexes of matroids are vertex decomposable and explores conditions under which these complexes are shellable, revealing new structural properties and relationships with lattice of flats.

Contribution

It establishes vertex decomposability of augmented Bergman complexes of matroids and characterizes shellability in relation to the lattice of flats.

Findings

Augmented Bergman complexes of matroids are vertex decomposable.

Shellability of augmented Bergman complexes is equivalent to shellability of the lattice of flats.

A shellable augmented Bergman complex may lack a basis-to-flag shelling order.

Abstract

We study the augmented Bergman complex of a closure operator on a finite set, which interpolates between the order complex of proper flats and the independence complex of the operator. In 2020, Braden, Huh, Matherne, Proudfoot, and Wang showed that augmented Bergman complexes of matroids are always gallery-connected, and recently Bullock, Kelley, Reiner, Ren, Shemy, Shen, Sun, Tao, and Zhang strengthened "gallery-connected" to "shellable" by providing two classes of shelling orders: "flag-to-basis" shellings and "basis-to-flag" shellings. We show that augmented Bergman complexes of matroids are vertex decomposable, a stronger property than shellable. We also prove that the augmented Bergman complex of any closure operator is shellable if and only if lattice of flats (that is, its non-augmented Bergman complex) is shellable. As a consequence, an augmented Bergman complex is shellable if and only if it admits a flag-to-basis shelling. Perhaps surprisingly, the same does not hold for basis-to-flag shellings: we describe a closure operator whose augmented Bergman complex is shellable, but has no shelling order with bases appearing first.