Eigenvalues of rotations and braids in spherical fusion categories

Abstract

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.

20 pages, 14 figures. Updated version to appear in J. Algebra

References in corpus (3)

• Modular data for the extended Haagerup subfactor
• Frobenius-Schur indicators for near-group and Haagerup-Izumi fusion categories
• Finite Linear Quotients of $\B_3$ of Low Dimension

Cited by in corpus (3)

• Fusion Rules for $\mathbb{Z}/2\mathbb{Z}$ Permutation Gauging
• A family of non-FSZ finite symplectic groups
• Skein-Theoretic Methods for Unitary Fusion Categories