Realizations of regular polytopes |
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| Authors: | Peter McMullen |
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| Institution: | 1. Department of Mathematics, University College London, Gower Street, WC1E 6BT, London, England
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| Abstract: | Let  be a finite regular incidence-polytope. A realization of is given by an imageV of its vertices under a mapping into some euclidean space, which is such that every element of the automorphism group  () of induces an isometry ofV. It is shown in this paper that the family of all possible realizations (up to congruence) of forms, in a natural way, a closed convex cone, which is also denoted by The dimensionr of is the number of equivalence classes under () of diagonals of , and is also the number of unions of double cosets * *  *–1* (  *), where * is the subgroup of () which fixes some given vertex of . The fine structure of corresponds to the irreducible orthogonal representations of (). IfG is such a representation, let its degree bed
G
, and let the subgroup ofG corresponding to * have a fixed space of dimensionw
G
. Then the relations
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