====== Cristina Ruiz (Binghamton) ======

====== A Stratification of the Middle-level MacPhersonian ======

===== Abstract for the Combinatorics Seminar 2005 March 31 =====

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 M<sub>1</sub> ≥ M<sub>2</sub> if there is a weak map from M<sub>1</sub> to M<sub>2</sub>. MacP(k,n) can be viewed as a combinatorial analog of the Grassmann manifold G(k,n) of k-planes in **R**<sup>n</sup>.

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 **R**<sup>n</sup>. 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.