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

TL;DR

This paper develops a unified, explicit method to compute the integral homology and Poincaré polynomials of real flag manifolds associated with classical and exceptional Lie algebras, addressing orientability issues.

Contribution

It introduces an algorithmic framework for homology calculations using Bruhat decomposition and normal forms of Weyl group elements, applicable to both classical and exceptional types.

Findings

Computed Poincaré polynomials for types B, C, D (up to n=7), F4, E6, E7.

Resolved sign determination in boundary operators via coordinate map degree analysis.

Addressed orientability of split real flag manifolds for exceptional Lie algebras.

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.