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We give a rigorous and self-consistent derivation of the elementary braidmatrices representing the exchanges of adjacent Ising anyons in the twoinequivalent representations of the Pfaffian quantum Hall states with even andodd number of Majorana fermions. To this end we use the distinct operatorproduct expansions of the chiral spin fields in the Neveu-Schwarz and Ramondsectors of the two-dimensional Ising conformal field theory. We find recursiverelations for the generators of the irreducible representations of the braidgroup B_{2n+2} in terms of those for B_{2n}, as well as explicit formulas foralmost all braid matrices for exchanges of Ising anyons. Finally we prove thatthe braid-group representations obtained from the multi-anyon Pfaffian wavefunctions are completely equivalent to the spinor representations of SO(2n+2)and give the equivalence matrices explicitly. This actually proves that thecorrelation functions of 2n chiral Ising spin fields sigma do indeed realizeone of the two inequivalent spinor representations of the rotation group SO(2n)as conjectured by Nayak and Wilczek.

Ultimate braid-group generators for coordinate exchanges of Ising anyons from the multi-anyon Pfaffian wave functions