Diagonal unitary and orthogonal symmetries in quantum theory

We analyze bipartite matrices and linear maps between matrix algebras, which are respectively, invariant covariant, under the diagonal unitary orthogonal groups' actions. By presenting an expansive list of examples from literature, includes notable entries like Diagonal Symmetric states Choi-type maps, we show that this class (and maps) encompasses a wide variety scenarios, thereby unifying their study. examine algebraic structure investigate different notions positivity through convex conic manifestations. In particular, generalize well-known cone completely positive to triplewise connect it separability relevant (or entanglement breaking property corresponding quantum channels). For provide explicit characterizations stated covariance in terms Kraus, Stinespring, Choi representations, systematically usual properties positivity, decomposability, complete like. also describe subspaces these use necessary sufficient conditions for associated states.