Let G be a connected reductive
linear algebraic group over the field of complex numbers, and B a fixed Borel
subgroup of G. The study of the homological properties of G∕B can be carried out by
two well-known methods. The first of these methods is due to A. Borel and
involves the identification of the cohomology ring of G∕B with the quotient
ring of the ring of polynomials on the Lie algebra h of the Cartan subgroup
H ⊂ G by the ideal generated by the W-invariant polynomials (where W is
the Weyl group of G). The second method is classical, and based on the
calculation of the homology with the aid of the partition of G∕B into cells, the
so-called Schubert cells. The correspondence between these approaches has
been studied in the paper by Bernstein, Gel’fand and Gel’fand, where in
the quotient ring of the polynomial ring figuring in Borel’s model of the
cohomology, the authors have found a symmetrical basis dual to the Schubert cells.
Moreover, they have given a formula (Intersection formula) which expresses the
intersection of any Schubert cell with a cell of codimension one. In the same
paper, the authors have also generalized these results, except the intersection
formula, to the case when B is replaced by an arbitrary parabolic subgroup
P ⊂ G.
On the other hand, for any parabolic subgroup P ⊂ G containing B, the
cohomology Gysin homomorphism of π : G∕B → G∕P has been studied by
Akyildiz and Carrell, where an explicit formula has been obtained between
the rings figuring in Borel’s model of the cohomologies. This formula also
enables one to obtain some of the results mentioned above for G∕P from
the corresponding results on G∕B. In this note, we consider the case where
G =GL(n + 1) and G∕P is the Grassmann manifold. By using the explicit
description of the Gysin homomorphism and the intersection formula given in the
cohomology ring of G∕B we obtain three main theorems of the symbolic formalism,
known as Schubert Calculus, concerning the cohomology ring structure of the
Grassmann manifold. Although there are several different approaches for
proving these theorems (see Kleiman and Laksov), it seems that none of
them uses the cohomology ring structure of GL(n + 1)∕B, where B is the
group of upper triangular matrices in GL(n + 1). We thus hope that this
alternative point of view may be used to understand the generalized Schubert
Calculus.
