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 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 C is also braided, these formulae yield the multiplicities 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.

A family of non-FSZ finite symplectic groups

“…Fusion categories are a topic of significant interest in mathematical physics, and one of the major invariants they are studied with are the higher Frobenius-Schur indicators [11,17]. These generalize the notion of Frobenius-Schur indicators for representations of finite groups, are algebraic integers in a cyclotomic field, and are akin to structure constants for certain algebraic objects within the category [1]. In the classical case of finite groups the indicators are in fact always integers, but this need not be true for more arbitrary categories.…”

Skein-Theoretic Methods for Unitary Fusion Categories

“…where the left and right diagrams respectively illustrate ϕ(f ) for an (anti)clockwise rotation and where 1 ≤ l ≤ min{m, n}. A further variant is studied in [35].…”