A correspondence between boundary coefficients of real flag manifolds and height of roots

TL;DR

This paper establishes a new formula linking boundary coefficients of real flag manifolds to root heights, simplifying calculations especially for type A, and explicitly determining some homology groups.

Contribution

It introduces a novel formula connecting cellular homology coefficients to root heights, providing explicit results for type A flag manifolds.

Findings

New formula for boundary coefficients in terms of root heights

Explicit expressions for first and second homology groups of type A flag manifolds

Simplified calculations for real flag manifolds of type A

Abstract

In this paper we prove a new formula for the coefficients of the cellular homology of real flag manifolds in terms of the height of certain roots. In particular, for flag manifolds of type A, we get a very simple formula for these coefficients and an explicit expression for the first and second homology groups with integer coefficients.