Integral homology of real isotropic and odd orthogonal Grassmannians

TL;DR

This paper provides a combinatorial formula for the boundary map coefficients in the integral homology of real isotropic and odd orthogonal Grassmannians, generalizing previous results and enabling computations of homology groups.

Contribution

It introduces a new combinatorial expression for boundary coefficients in these Grassmannians, extending known formulas and offering tools for homology calculations.

Findings

Derived a combinatorial formula for boundary map coefficients

Established an orientability criterion for these Grassmannians

Computed low-dimensional homology groups

Abstract

We obtain a combinatorial expression for the coefficients of the boundary map of real isotropic and odd orthogonal Grassmannians providing a natural generalization of the formulas already obtained for Lagrangian and maximal isotropic Grassmannians. The results are given in terms of the classification into four types of covering pairs among the Schubert cells when identified with signed $k$-Grassmannian permutations. It turns out that these coefficients only depend on the positions changed over each pair of permutations. As an application, we give an orientability criterion, exhibit a symmetry of these coefficients and, compute low-dimensional homology groups.