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*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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