Homogeneous Einstein metrics on generalized flag manifolds with five isotropy summands

TL;DR

This paper classifies and constructs invariant Einstein metrics on certain generalized flag manifolds with five isotropy summands, introducing a new technique based on Riemannian submersions and polynomial system analysis.

Contribution

It develops a novel method using flag manifold fibrations and Gr"obner bases to find Einstein metrics on complex homogeneous spaces with five isotropy components.

Findings

Explicit Einstein metrics on specific flag manifolds E_6 and E_7.

Proof of existence of multiple non K"ahler-Einstein metrics on certain SO(2 ext{ exttwosuperior})-type manifolds.

Determination of the exact number of Einstein metrics for small parameters.

Abstract

We construct the homogeneous Einstein equation for generalized flag manifolds $G/K$ of a compact simple Lie group $G$ whose isotropy representation decomposes into five inequivalent irreducible $\Ad(K)$-submodules. To this end we apply a new technique which is based on a fibration of a flag manifold over another flag manifold and the theory of Riemannian submersions. We classify all generalized flag manifolds with five isotropy summands, and we use Gr\"obner bases to study the corresponding polynomial systems for the Einstein equation. For the generalized flag manifolds E_6/(SU(4) x SU(2) x U (1) x U (1)) and E_7/(U(1) x U(6)) we find explicitely all invariant Einstein metrics up to isometry. For the generalized flag manifolds SO(2\ell +1)/(U(1) x U (p) x SO(2(\ell -p-1)+1)) and SO(2\ell)/(U(1) x U (p) x SO(2(\ell -p-1))) we prove existence of at least two non K\"ahler-Einstein metrics. For small values of $\ell$ and $p$ we give the precise number of invariant Einstein metrics.