**Problem of the Week**

**Math Club**

**BUGCAT 2020**

**Zassenhaus Conference**

**Hilton Memorial Lecture**

seminars:comb:abstract.200503rui

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_{1} ≥ M_{2} if there is a weak map from M_{1} to M_{2}. MacP(k,n) can be viewed as a combinatorial analog of the Grassmann manifold G(k,n) of k-planes in **R**^{n}.

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**^{n}. 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.

seminars/comb/abstract.200503rui.txt · Last modified: 2020/01/29 14:03 (external edit)

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