Degeneracy and non-Abelian statistics

A non-Abelian anyon can only occur in the presence of ground -state degeneracy in the plane. It is conceivable that for some strange anyon with quantum dimension > 1 that the resulting representations of all n-strand braid groups B, are overall phases, even though the ground -state manifolds for n such anyons in the plane are in general Hilbert spaces of dimensions > 1. We observe that degeneracy is all that is needed: For an anyon with quantum dimension > 1 the non-Abelian statistics cannot all be overall phases on the degeneracy ground -state manifold. Therefore, degeneracy implies non-Abelian statistics, which justifies defining a non-Abelian anyon as one with quantum dimension > 1. Since non-Abelian statistics presumes degeneracy, degeneracy is more fundamental than non-Abelian statistics.