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