  We give formulae for the multiplicities of eigenvalues of generalized
rotation operators in terms of generalized Frobenius-Schur indicators in a
semisimple spherical tensor category $\mathcal{C}$. In particular, this implies
that the entire collection of rotation eigenvalues for a fusion category can be
computed from the fusion rules and the traces of rotation at finitely many
tensor powers. We also establish a rigidity property for FS indicators of
fusion categories with a given fusion ring via Jones's theory of planar
algebras. If $\mathcal{C}$ is also braided, these formulae yield the
multiplicities of eigenvalues for a large class of braids in the associated
braid group representations. When $\mathcal{C}$ is modular, this allows one to
determine the eigenvalues and multiplicities of braids in terms of just the $S$
and $T$ matrices.
