Integral homology of real isotropic and odd orthogonal Grassmannians

Abstract

We obtain a combinatorial expression for the boundary map coefficients of real isotropic and odd orthogonal Grassmannians. It provides a natural generalization of the known formulas for Lagrangian and maximal isotropic Grassmannians. The results derive from the classification of Schubert cells into four types of covering pairs when identified with signed $k$-Grassmannian permutations. Our formulas show that the coefficients depend on the changed positions for each permutation pair type. We apply this to obtain an orientability criterion and compute the first and second homology groups for these Grassmannians. Furthermore, we exhibit an apparent symmetry of the boundary map coefficients.