Orbit-entropy cones and extremal pairwise orbit-entropy inequalities

The notion of orbit-entropy cone is introduced. Specifically, orbit-entropy cone equation is the projection of equation induced by G, where equation is the closure of entropy region for n random variables and G is a permutation group over {0; 1;...; n-1}. For symmetric group Sn (with arbitrary n) and cyclic group Cn (with n ≤ 5), the associated orbit-entropy cones are shown to be characterized by the Shannon type inequalities. Moreover, the extremal pairwise relationship between orbit-entropies is determined completely for partitioned symmetric groups and partially for cyclic groups.