Indistinguishable Chargeon-Fluxion Pairs in the Quantum Double of Finite Groups

We consider the category of finite dimensional representations of the quantum double of a finite group as a modular tensor category. We study auto-equivalences of this category whose induced permutations on the set of simple objects (particles) are of the special form of PJ, where J sends every particle to its charge conjugation and P is a transposition of a chargeon-fluxion pair. We prove that if the underlying group is the semidirect product of the additive and multiplicative groups of a finite field, then such an auto-equivalence exists. In particular, we show that for S_3 (the permutation group over three letters) there is a chargeon and a fluxion which are not distinguishable. Conversely, by considering such permutations as modular invariants, we show that a transposition of a chargeon-fluxion pair forms a modular invariant if and only if the corresponding group is isomorphic to the semidirect product of the additive and multiplicative groups of a finite near-field.