Einstein equations for invariant metrics on flag spaces and their Newton polytopes

Abstract:

This paper deals with the number of complex invariant Einstein metrics on flag spaces in the case when the isotropy representation has a simple spectrum. The author has previously showed that this number does not exceed the volume of the Newton polytope of the Einstein equation (in this case, this is a rational system of equations), which coincides with the Newton polytope of the scalar curvature function. The equality is attained precisely when that function has no singular points on the faces of the polytope, which is the case for “pyramidal faces”. This paper studies non-pyramidal faces. They are classified with the aid of ternary symmetric relations (which determine the Newton polytope) in the $T$-root system (the restriction of the root system of the Lie algebra of the group to the center of the isotropy subalgebra). The classification is mainly done by computer-assisted calculations for classical and exceptional groups in the case when the number of irreducible components does not exceed 10 (and, in some cases, 15).