  We study the Rees valuations and rational powers of several classes of
invariant ideals, providing explicit descriptions in terms of certain
polyhedra. This allows us to show that a version of Musta\c{t}\u{a}-Takagi's
summation formula for rational powers holds for these ideals. Moreover, for
arbitrary ideals over the complex numbers, we prove a weaker version of this
formula that holds for high enough rational numbers. Our methods lead to a
definition of rational powers of ideal sheaves in normal complex varieties
inspired by work of Lejeune-Jalabert-Teissier and Lazarsfeld.
