In this work, the authors revisit the determination of the boundary map coefficients for the cellular homology of real flag manifolds, a problem equivalent to finding the incidence coefficients of the differential map for the cohomology. They prove a new formula for these coefficients with respect to the height of certain roots.
A generalized flag manifold \(\mathbb F \) is a homogeneous space \(G/P\), where \(G\) is a real noncompact semisimple Lie group and \(P\) is a parabolic subgroup. It admits a cellular decomposition called Bruhat decomposition, where the cells are the Schubert cells and parametrized by the Weyl group \(W\). There is the Bruhat Chevalley order on elements of the Weyl group. In this case, there is a root \(\beta\) such that \(w = s_{\beta}\cdot w'\). In both [R. R. Kocherlakota, Adv. Math. 110, No. 1, 1–46 (1995; Zbl 0832.22020)] and [L. Rabelo and L. A. B. San Martin, Indag. Math., New Ser. 30, No. 5, 745–772 (2019; Zbl 1426.57052)], the authors summarized how to compute the coefficient \(c(w,w')\).
The papers [L. Rabelo, Adv. Geom. 16, No. 3, 361–379 (2016; Zbl 1414.57018)] and [J. Lambert and L. Rabelo, Australas. J. Comb. 75, Part 1, 73–95 (2019; Zbl 1429.05005)] apply this procedure in the context of the Isotropic Grassmannians and the results obtained (for instance, see [J. Lambert and L. Rabelo, “Integral homology of real isotropic and odd orthogonal Grassmannians”, Preprint, arXiv:1604.02177, to appear in Osaka J. Math.], Theorem 3.12) suggest a formula for the coefficients in terms of the height of some root.
Overall they prove a new formula for the cellular homology coefficients of real flag manifolds in terms of the height of certain roots. For real flag manifolds of type \(A\), they get simple expressions for the coefficients that allow us to compute the first and second integral homology groups exhibiting their generators.