The augmented external activity complex of a matroid

For a matroid, we define a new simplicial complex whose facets are indexed by its independent sets. This complex contains the external activity complex as a subcomplex. We call our complex the augmented external activity complex since its definition is motivated by the recently defined augmented tautological classes of matroids. We prove that our complex is shellable and show that our shelling satisfies the stronger property of being an $H$-shelling. This explicates our result that the $h$-vector of our complex is the $f$-vector of the independence complex. We also define an augmented no broken circuit complex, which contains the usual no broken circuit complex as a subcomplex. We prove its shellability and show that our shelling is also an $H$-shelling. The $h$-vector of this complex is the $f$-vector of the no broken circuit complex.