Topology of augmented Bergman complexes

TL;DR

This paper proves the shellability of the augmented Bergman complex of a matroid, explores its homotopy type, and reveals a simple automorphism group action on its homology, extending to more general closures.

Contribution

It establishes the shellability of the augmented Bergman complex using two shellings and links its homotopy type to a known basis counting formula, also describing its automorphism group action.

Findings

Augmented Bergman complex is shellable via two shellings.

Homotopy type relates to a basis counting convolution formula.

Automorphism group acts simply on the homology.

Abstract

The augmented Bergman complex of a matroid is a simplicial complex introduced recently in work of Braden, Huh, Matherne, Proudfoot and Wang. It may be viewed as a hybrid of two well-studied pure shellable simplicial complexes associated to matroids: the independent set complex and Bergman complex. It is shown here that the augmented Bergman complex is also shellable, via two different families of shelling orders. Furthermore, comparing the description of its homotopy type induced from the two shellings re-interprets a known convolution formula counting bases of the matroid. The representation of the automorphism group of the matroid on the homology of the augmented Bergman complex turns out to have a surprisingly simple description. This last fact is generalized to closures beyond those coming from a matroid.