Homogeneous Einstein metrics on flag manifolds

It is known that a flag manifold admits a Käahler–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.