Yusuke Sakane
Homogeneous Einstein metrics on flag manifolds
(Lobachevskii Journal of Mathematics, Vol.4, pp.71-87)

It is known that a flag manifold admits a K\"ahler-Einstein metric. We investigate $K$-invariant Einstein metrics on a flag manifold $M = K/T$ which is not K\"ahler-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 Gr\"obner 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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