Rational powers, invariant ideals, and the summation formula

We provide explicit descriptions for the rational powers and Rees valuations of several classes of ideals invariant under natural actions of tori and products of general linear groups, in terms of polyhedra and lattice points. This allows us to show that a version of Musta\c{t}\u{a}-Takagi's summation formula for multiplier ideals also holds for the rational powers of these ideals. Moreover, for arbitrary ideals in normal domains that are finitely generated over algebraically closed fields, we prove a weaker version of this formula that holds for sufficiently large rational numbers.