Singular and de Rham cohomology for the Grassmannian

Abstract

We compute the Poincar e polynomial for the complex Grassmannian using de Rham cohomology. We also construct a CW complex on the Grassmannian using Schubert cells, and then we use these cells to construct a basis for the singular cohomology. We give an algorithm for calculating the number of cells, and use this to compare the basis in singular cohomology with the Poincar e polynomial from de Rham cohomology. We also explore Schubert calculus and the connection between singular cohomology on the complex Grassmannian and the possible triples of eigenvalues to Hermitian matrices A+B = C, and give a brief discussion on if and how cohomologies can be used in the case of real skew-symmetric matrices.