Geodesic
Einstein equations for invariant metrics on flag spaces and their Newton polytopes
Trudy Moskovskogo matematičeskogo obŝestva, Trudy Moskovskogo Matematicheskogo Obshchestva, Tome 75 (2014) no. 1, pp. 25-91.

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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). 
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M. M. Graev. Einstein equations for invariant metrics on flag spaces and their Newton polytopes. Trudy Moskovskogo matematičeskogo obŝestva, Trudy Moskovskogo Matematicheskogo Obshchestva, Tome 75 (2014) no. 1, pp. 25-91. http://geodesic.mathdoc.fr/item/MMO_2014_75_1_a2/

[1] Alekseevskii D. V., Perelomov A. M., “Invariantnye metriki Kelera–Einshteina na kompaktnykh odnorodnykh prostranstvakh”, Funkts. analiz i ego pril., 20:3 (1986), 1–16

[2] Besse A., Mnogoobraziya Einshteina, V 2 t., Mir, M., 1990 | MR | Zbl

[3] Bernshtein D. N., “Chislo kornei sistemy uravnenii”, Funkts. analiz i ego pril., 9:3 (1975), 1–4

[4] Danilov V. I., “Geometriya toricheskikh mnogoobrazii”, UMN, 33:2 (1978), 85–135

[5] Gelfand I. M., Kapranov M. M., Zelevinsky A. V., Discriminants, resultants, and multidimensional determinants, Birkhäuser, Boston, 1994 | MR

[6] Fulton W., Introduction to toric varieties, Princeton University Press, 1993 | MR | Zbl

[7] Vinberg E. B., Onischik A. L., Seminar po gruppam Li i algebraicheskim gruppam, Nauka, M., 1988 | MR

[8] Vinberg E. B., Gorbatsevich V. V., Onischik A. L., “Stroenie grupp i algebr Li”, Itogi nauki i tekhn. Sovrem. probl. matem. Fundam. napravleniya, 41, VINITI, M., 1990, 5–253

[9] Graev M. M., “On the number of invariant Einstein metrics on a compact homogeneous space, Newton polytopes and contractions of Lie algebras”, International Journal of Geometric Methods in Modern Physics, 3:5–6 (2006), 1047–1075 | DOI | MR | Zbl

[10] Graev M. M., “Chislo invariantnykh metrik Einshteina v odnorodnom prostranstve, mnogogrannik Nyutona i szhatiya algebry Li”, Izvestiya RAN. Ser. matem., 72:2 (2007), 29–88 | DOI

[11] Graev M. M., On invariant Einstein metrics on Kähler homogeneous spaces $SU_4/T^3$, $G_2/T^2$, $E_6/T^2(A_2)^2$, $E_7/T^2A_5$, $E_8/T^2E_6$, $F_4/T^2A_2$, arXiv: 1111.0639 | MR

[12] Graev M. M., On the compactness of the set of invariant Einstein metrics, arXiv: 1207.3034v2 | MR