Non-Abelian statistics of vortices with multiple Majorana fermions

We consider the exchange statistics of vortices, each of which traps an odd number (N) of Majorana fermions. We assume that the fermions in a vortex transform in the vector representation of the SO(N) group. Exchange of two vortices turns out to be non-Abelian, and the corresponding operator is further decomposed into two parts: a part that is essentially equivalent to the exchange operator of vortices having a single Majorana fermion in each vortex, and a part representing the Coxeter group. Similar decomposition was already found in the case with N=3, and the result shown here is a generalization to the case with an arbitrary odd N. We can obtain the matrix representation of the exchange operators in the Hilbert space that is constructed by using Dirac fermions nonlocally defined by Majorana fermions trapped in separated vortices. We also show that the decomposition of the exchange operator implies tensor-product structure in its matrix representation.