  Peter McMullen has developed a theory of realizations of abstract regular
polytopes, and has shown that the realizations up to congruence form a pointed
convex cone which is the direct product of certain irreducible subcones. We
show that each of these subcones is isomorphic to a set of positive
semi-definite hermitian matrices of dimension $m$ over either the real numbers,
the complex numbers or the quaternions. In particular, we correct an erroneous
computation of the dimension of these subcones by McMullen and Monson. We show
that the automorphism group of an abstract regular polytope can have an
irreducible character $\chi$ with $\chi\neq \overline{\chi}$ and with
arbitrarily large essential Wythoff dimension. This gives counterexamples to a
result of Herman and Monson, which was derived from the erroneous computation
mentioned before.
  We also discuss a relation between cosine vectors of certain pure
realizations and the spherical functions appearing in the theory of Gelfand
pairs.
