# 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.