The External Activity Complex of a Pair of Matroids

I will introduce the external activity complex of an ordered pair of matroids on the same ground set. This is the combinatorial analogue of the Schubert variety of an ordered pair of linear subspaces of a fixed space, which is also new. These constructions generalize and extend the well-studied notion of external activity for a matroid basis, and the Schubert variety of a single linear space, which occur in the bivariate setting when one of the matroids is uniform of rank 1. Certain Betti numbers of the external activity complex of a pair determine the Euler characteristic of various (non-positive) tautological classes of matroids, and this can be used to give a positive formula for Speyer's matroid invariant $\(\omega(M)\)$. This is on-going work with Alex Fink.