Homogeneous Einstein metrics on \(G_{2}/T\) (Q2839319)

The authors investigate homogeneous Einstein metrics on \(G_2/T\), where \(G_2\) is the exceptional simple compact Lie group and \(T\) is a maximal torus in \(G_2\). This space belongs to the class of full flag manifolds.NEWLINENEWLINEIn the case of homogeneous spaces the Einstein equation reduces to a system of algebraic equations. For some cases positive solutions (which give us Einstein metrics) can been obtained explicitly. The main result of this paper is the following:NEWLINENEWLINETheorem A. The full flag manifold \(G_2/T\) admits exactly three \(G_2\)-invariant Einstein metrics (up to isometry). There is a unique Kähler-Einstein metric (up to a scalar) and the other two are not Kähler. NEWLINENEWLINENEWLINE The approximate values of these invariant metrics are given in this paper too (Theorem 4.1).NEWLINENEWLINEThe proof of this theorem is based, in particular, on solving the system of algebraic equations mentioned above. To solve this system the authors consider two cases -- A and B. In the case A they find (by some computer manipulations not given in detail) a Gröbner basis \(p_1,p_2,p_3\). Some coefficients of these polynomials \(p_i\) are integers with approximately 50(!)~digits. The proof of case A is done by numerically solving the equation \(p_1=0\) (of degree 14), and then by solving numerically the equations \(p_1 =0, p_2=0\). The case B is considered analogously. So, this proof is very difficult to check by hand.NEWLINENEWLINEAs a consequence of Theorem A, the space \(G_2/T\) gives the first known example of an exceptional full flag manifold which admits a non-Kähler and not normal homogeneous Einstein metric. The isotropy representation of \(G_2/T\) decomposes into six inequivalent irreducible submodules. The isotropy representation of the full flag manifolds corresponding to the other exceptional Lie groups \(F_4, E_6, E_7, E_8\) decomposes into much more irreducible summands, so searching for non-Kähler and not normal homogeneous Einstein metrics in these cases seems to be a very difficult problem.