Isotropy summands and Einstein Equation of Invariant Metrics on Classical Flag Manifolds

TL;DR

This paper derives the algebraic Einstein equations for invariant metrics on classical flag manifolds, determines the number of isotropy summands, and analyzes properties of t-roots for these spaces.

Contribution

It provides a comprehensive description of Einstein equations and isotropy summands for all classical Lie group flag manifolds, including properties of t-roots.

Findings

Algebraic system for Einstein metrics on classical flag manifolds derived

Number of isotropy summands determined for these manifolds

Properties of t-roots for types B_n, C_n, D_n analyzed

Abstract

It is well known that the Einstein equation on a Riemannian flag manifold $(G/K,g)$ reduces to a algebraic system, if $g$ is a $G$-invariant metric. In this paper we described this system for all flag manifolds of a classical Lie group. We also determined the number of isotropy summands for all of these spaces and proved certain properties of the set of t-roots for flag manifolds of type $B_n$, $C_n$ and $D_n$.