Invariant embeddings and weighted permutations.

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