A geometric interpretation of an equality by Sylvester Original Research Article

In this paper, we will show that a classical theorem of topology is, in fact, a combinatorial one, holding in all projective spaces. In the finite case, that is in the case of Galois geometries, this result enables us to obtain many equations between parameters. The easier ones, relating to the complete cell decomposition of Grassmann varieties, are the classical identities introduced by Sylvester in the study of the theory of the partitions. So we start by recalling the Sylvester equality; in Section 2 we shall define Schubert cells and related theorems in the abstract case; finally, in Section 3 we shall consider Galois projective spaces.