The detailed study of the Grassmannians uses a decomposition into subsets called Schubert cells, which were first applied in enumerative geometry. The Schubert cells for Gr(r, n) are defined in terms of an auxiliary flag: take subspaces V1, V2, ..., Vr, with Vi contained in Vi + 1. Then we consider the corresponding subset of Gr(r, n), consisting of the W having intersection with Vi of dimension at least i, for i = 1 to r. The manipulation of Schubert cells is Schubert calculus.
Here is an example of the technique. Consider the problem of determining the Euler characteristic of the Grassmannian of r-dimensional subspaces of Rn. Fix a one-dimensional subspace R ⊂ Rn and consider the partition of Gr(r, n) into those r-dimensional subspaces of Rn that contain R and those that do not. The former is Gr(r−1, n−1) and the latter is a r-dimensional vector bundle over Gr(r, n−1). This gives recursive formulas:
where by design . If one solves this recurrence relation, one gets the formula: if and only if n is even and r is odd. Otherwise:
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