The MacPhersonian MacP(k,n) is the partially ordered set of all oriented matroids of rank k on the ground set {1, 2, ..., n}, ordered by M1 ≥ M2 if there is a weak map from M1 to M2. MacP(k,n) can be viewed as a combinatorial analog of the Grassmann manifold G(k,n) of k-planes in Rn.
The Grassmannian G(k,n) has a type of cell decomposition called a Schubert cell decomposition. To define the cells we need to fix some subspaces of Rn. It is known that for a special Schubert cell decomposition of G(k,n), we can give an explicit combinatorial definition of ``cells'' for a ``cell decomposition'' of MacP(k,n). This combinatorial analog of a Schubert cell decomposition of G(k,n) is called a Schubert stratification of MacP(k,n).
In studying spectral structures on MacP(k, infinity), I found that another stratification of MacP(n, 2n), based on a different Schubert cell decomposition of G(n, 2n), looks promising. I will show the ideas behind this work in progress.