Title  Homogeneous Einstein metrics on generalized flag manifolds with five isotropy summands. 
Author(s)  Andreas Arvanitoyeorgos, Ioannis Chrysikos, Yusuke Sakane 
Type  Article in Journal 
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 such space and the theory of Riemannian submersions. We classify all generalized flag manifolds with five isotropy summands, and we use Gröbner bases to study the corresponding polynomial systems for the Einstein equation. For the generalized flag manifolds E6/(SU(4) × SU(2) × U(1) × U(1)) and E7/(U(1) × U(6)) we find explicitly all invariant Einstein metrics up to isometry. For the generalized flag manifolds SO(2ℓ + 1)/(U(1) × U(p) × SO(2(ℓ  p  1) + 1)) and SO(2ℓ)/(U(1) × U(p) × SO(2(ℓ  p  1))) we prove existence of at least two nonKähler–Einstein metrics. For small values of ℓ and p we give the precise number of invariant Einstein metrics.

Keywords  Homogeneous space; Einstein metric; Riemannian submersion; generalized flag manifold; isotropy representation; Weyl group 
ISSN  0129167X; 17936519/e 
URL 
http://www.worldscientific.com/doi/abs/10.1142/S0129167X13500778 
Language  English 
Journal  Int. J. Math. 
Volume  24 
Number  10 
Pages  52 
Publisher  World Scientific, Singapore 
Year  2013 
Edition  0 
Translation 
No 
Refereed 
No 