Equivariant Relative Submajorization

We study a generalization of relative submajorization that compares pairs positive operators on representation spaces some fixed group. A pair equivariantly relatively submajorizes another if there is an equivariant subnormalized channel takes the components first to satisfying similar positivity constraints as in definition submajorization. In context resource theory approach thermodynamics, this allows one transformations by Gibbs-preserving maps are addition time-translation symmetric. find sufficient condition for existence catalytic and characterization asymptotic relaxation relation. For classical certain quantum terms explicit monotone quantities related sandwiched Rényi divergences. general case relevant given only implicitly. Nevertheless, we large collection monotones provide necessary conditions or transformations. When applied symmetric maps, these give rise second laws constrain state allowed thermal operations even presence catalysts.