  We prove a variety of results describing the possible diagonals of tuples of
commuting hermitian operators in type $II_1$ factors. These results are
generalisations of the classical Schur-Horn theorem to the infinite
dimensional, multivariable setting. Our description of these possible diagonals
uses a natural generalisation of the classical notion of majorization to the
multivariable setting. In the special case when both the given tuple and the
desired diagonal have finite joint spectrum, our results are complete. When the
tuples do not have finite joint spectrum, we are able to prove strong
approximate results. Unlike the single variable case, the multivariable case
presents several surprises and we point out obstructions to extending our
complete description in the finite spectrum case to the general case. We also
discuss the problem of characterizing diagonals of commuting tuples in
$\mathcal{B}(\mathcal{H})$ and give approximate characterizations in this case
as well.
