Orbit-Entropy Cones and Extremal Pairwise Orbit-Entropy Inequalities

The notion of orbit-entropy cone is introduced. Specifically, orbit-entropy cone $P_{G}\overline{\Gamma}_{n}^{\ast}$ is the projection of $\overline{\Gamma}_{n}^{\ast}$ induced by $G$, where $\overline{\Gamma}_{n}^{\ast}$ is the closure of entropy region for $n$ random variables and $G$ is a permutation group over $\{0,1,\cdots,n-1\}$. For symmetric group $S_{n}$ (with arbitrary n) and cyclic group $C_{n}$ (with $n\leq 5$), 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.