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Betti numbers of Springer fibers in type A
Let u be a nilpotent endomorphism of a finite dimensional vector space.
The set of all u-stable complete flags is a projective algebraic variety,
called Springer fiber. We compute the Betti numbers of this variety. To do
this, we construct a cell decomposition. The cells are parameterized by a
set of tableaux, and the codimension of the cells is given by an inversion
number. This makes our decomposition suited for the calculation of Betti
numbers. When u is zero, the Springer fiber is the whole flag variety and
our decomposition coincides with the classical decomposition into Schubert
cells.
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