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Realizations of abstract regular polytopes from a representation theoretic view

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 $χ$ with $χ\neq \overlineχ$ 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.