Integral Homology and Poincar\'e Polynomials of classical and exceptional Real Flag Manifolds

Abstract

This paper computes the integral homology of real flag manifolds associated with split real forms of classical and exceptional semisimple Lie algebras. Using the cellular homology provided by the Bruhat decomposition, we introduce a unified framework to systematically determine the coefficients of the boundary operator, explicitly resolving the issue of calculating their signs. This is achieved by computing the degree of change of coordinate maps between different reduced decompositions of Weyl group elements, analyzing commutation and braid relations through Lie bracket computations and exponential identities. By adopting the normal form of Weyl group elements as a canonical choice for reduced decompositions, we establish an explicit algorithmic implementation for these homology computations. As a direct application, we derive the Poincar\'e polynomials for the classical types $B_n, C_n$, and $D_n$ for $n \leqslant 7$, and for the exceptional types $F_4, E_6$, and $E_7$. With the aid of these polynomials, we address the question of the orientability of split real flag manifolds of exceptional Lie algebras.