Cases of equality for certain inequalities involving a G-doubly stochastic map

We study inner product inequalities for a G-doubly stochastic map with respect to a group induced cone ordering. We give necessary and sufficient conditions for the inequalities to become equalities. A special condition is obtained if a Birkhoff type theorem holds for a finite reflection group related to the ordering. The conditions are applied to derive some classical and new results on equality cases for majorization inequalities connected with certain group actions.