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

翻译：我们研究相对次多数的概括化,比较一些固定组群代表空间的正数操作者。一对相对相对相对的次分类化,如果有一个等同的次整化渠道,取第一对的成分,对一对满足相对次分化定义中类似现实性限制的对一对,则对一对相对次整化。在对热动力学的资源理论方法中,这种概括化使得人们可以研究Gibbs-保藏图的转换,这些转换加上时间翻译的对称。我们找到了催化转换和关系无症状放松特征的充分条件。对于古典和某些量配对来说,定性是用与配对的量子R\'enyi差异有关的明确的单体数量来表示的。在一般量的情况下,有关数量只是隐含的。然而,我们发现大量单体元素的收集,为无症状或催化变换提供了必要的条件。当应用时间转换对称地图时,这些特征产生了限制国家变换的第二法律,即使是在热力操作中也允许存在。