Matroids vs. Shifted simplicial complexes

The families of matroid independence complexes and pure shifted simplicial complexes are two remarkable classes of simplicial complexes that share several structural properties, but for which theorems are typically proved in different ways. In a groundbreaking paper Kook, Reiner and Stanton [4] proved that matroid complexes are Laplacian integral. Afterwards Duval and Reiner [1] proved the analogue of this result for shifted complexes using completely different techniques. Finally, Duval [2] proved that the Laplacians of both families of complexes satisfy the same (relative) deletion contraction recurrence. It is then reasonable to ask whether there is a natural class of simplicial complexes that bridges these two theories. We propose an approach that aims to construct such a class of simplicial complexes. The idea is to relax the classical cryptomorphic matroid axioms to obtain various classes of ordered complexes that contain both the pure shifted complexes and matroids. Each of these classes inherits some nice properties of matroids and, surprisingly, the complexes in the intersection of the mentioned classes enjoy enjoy even more analogue properties of classical matroid theory. We describe two such classes here: the first one is obtained by relaxing the exchange axiom, while the second one is obtained by relaxing the circuit axiom. We then proceed to study properties in the intersection of these two classes: they admit natural shelling orders, a notion of internal and external activities, and a well behaved theory of Tutte-Grothendieck invariants with a corresponding Tutte polynomial. In what follows, ∆ is a pure finite (d− 1)-simplicial complex. We distinguish between the ground set of ∆, which we call E, and the set of vertices which we call V . It is always true that V ⊆ E, but it is possible to have some e ∈ E for which {e} / ∈ ∆. We also assume that E comes endowed with a total order.