It has been a long-standing conjecture that the existence of "canonical" Kahler metrics would be related to an appropriate notion of "stability" in the sense of Mumford's Geometric Invariant Theory (GIT). In this talk, we will discuss about the relationships between Chow stability and weight polytope, building upon work of Gelfand-Kapranov-Zelevinsky. Later on, we will apply this combinatorial characterization to an example of Nill-Paffenholz, that is, 7-dimensional non-symmetric Kahler-Einstein toric Fano manifold. This is joint work with H.Ono and Y.Sano.