Geodesic
Homogeneous Einstein metrics on flag manifolds
Lobachevskii journal of mathematics, Tome 4 (1999), pp. 71-87.

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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. 
@article{LJM_1999_4_a3,
     author = {Yu. Sakane},
     title = {Homogeneous {Einstein} metrics on flag manifolds},
     journal = {Lobachevskii journal of mathematics},
     pages = {71--87},
     publisher = {mathdoc},
     volume = {4},
     year = {1999},
     language = {en},
     url = {http://geodesic.mathdoc.fr/item/LJM_1999_4_a3/}
}
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Yu. Sakane. Homogeneous Einstein metrics on flag manifolds. Lobachevskii journal of mathematics, Tome 4 (1999), pp. 71-87. http://geodesic.mathdoc.fr/item/LJM_1999_4_a3/