Eigenvalues of rotations and braids in spherical fusion categories

 In this talk, we will discuss generalized rotation operators in (spherical) fusion categories, and present formulas for the multiplicities of their eigenvalues in terms of generalized Frobenius-Schur indicators.  As an application, we will show when $C$ is modular how the above formulas also yield multiplicities of eigenvalues for a large class braids in braid group representations associated to $C$ in terms of just the modular data.