Information inequalities and finite groups: an overview

An entropic vector is a 2ⁿ - 1 dimensional vector collecting all the possible joint entropies of n discrete jointly distributed random variables. The region Γ*_n of entropic vectors plays a significant role in network information theory, since it is known that the capacity of large classes of networks can be computed by optimising a linear function over Γ*_n, under linear constraints, where n is the number of variables involved in a given network. However so far only Γ*₂ and Γ*₃ are known. One approach to study Γ*_n is to identify smaller regions that serve as inner bounds, and to look for entropic vectors violating inequalities characterising these regions. For example for n = 4, the inequality of interest is the so-called Ingleton inequality. We give an overview of recent work studying entropic vectors using a group theoretic approach. We recall Chan´s technique of constructing random variables from groups and the corresponding notion of (abelian) group representable entropic vectors. We review different works on groups yielding violations of linear rank inequalities, and discuss the classification of finite groups based on the entropic vector that they yield.