Yusuke Sakane

Homogeneous Einstein metrics on flag manifolds

(Lobachevskii Journal of Mathematics, Volume IV)

It is known that a flag manifold admits a Kähler-Einstein metric. We investigate K-invariant Einstein metrics on a flag manifold M = K/T which is not Kähler-Einstein. This problem has been studied by Alekseevsky and Arvanitoyeorgos in case of generalized flag manifolds. We give an explicit expression of Ricci tensor of a flag manifold K/T for the case of a classical simple Lie group and we present more new K-invariant Einstein metrics on a flag manifold K/T. We compute a Gröbner basis for a system of polynomials of multi-variables and show the existence of positive solutions for a system of algebraic equations to prove the existence of K-invariant Einstein metrics.
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