Eigenvalues of Rotations and Braids in Spherical Fusion Categories

We give formulas for the multiplicities of eigenvalues of generalized rotation operators in terms of generalized Frobenius-Schur indicators in a semi-simple spherical tensor category C. In particular, this implies for a finite depth planar algebra, the entire collection of rotation eigenvalues can be computed from the fusion rules and the traces of rotation at finitely many depths. If C is also braided, these formulas yield the multplicities of eigenvalues for a large class of braids in the associated braid group representations. When C is modular, this allows one to determine the eigenvalues and multiplicities of braids in terms of just the S and T matrices.