Homogeneous Einstein Metrics on Certain Generalized Flag Manifolds with Six Isotropy Summands

Abstract

For a generalized flag manifold M = G/K of a compact connected simple Lie group G whose isotropy representation decomposes into more than five isotropy summands, there are only a few results about the homogeneous Einstein metrics on M. Finding the invariant Einstein metrics on generalized flag manifolds, there are two difficulties. One is computing the non-zero structure constants, the other is computing the Grobner basis of the system of Einstein equations. In this paper, we give a method (Theorem A) which can be used to calculate structure constants of generalized flag manifolds with any number of isotropy summands. In this direction we present invariant Einstein metrics on some flag manifolds of exceptional groups with six isotropy summands.