Invariant embeddings and weighted permutations

We prove that for any fixed unitary matrix U U, any abelian self-adjoint algebra of matrices that is invariant under conjugation by U U can be embedded into a maximal abelian self-adjoint algebra that is still invariant under conjugation by U U. We use this result to analyse the structure of matrices A A for which A ∗ A A^*A commutes with A A ∗ AA^*, and to characterize matrices that are unitarily equivalent to weighted permutations.