Realizations of regular polytopes

Summary

Let ℐ be a finite regular incidence-polytope. A realization of ℐ 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 Γ(ℐ) of ℐ induces an isometry ofV. It is shown in this paper that the family of all possible realizations (up to congruence) of ℐ forms, in a natural way, a closed convex cone, which is also denoted by ℐ The dimensionr of ℐ is the number of equivalence classes under Γ(ℐ) of diagonals of ℐ, and is also the number of unions of double cosets Γ^*σΓ^* ∪ Γ^*σ⁻¹Γ^* (σ ∉ Γ^*), where Γ^* is the subgroup of Γ(ℐ) which fixes some given vertex of ℐ. The fine structure of ℐ corresponds to the irreducible orthogonal representations of Γ(ℐ). IfG is such a representation, let its degree bed _G , and let the subgroup ofG corresponding to Γ^* have a fixed space of dimensionw _G . Then the relations

$$\begin{array}{l} \Sigma _G w_G d_G = \upsilon - 1, \\ \Sigma _G {\textstyle{1 \over 2}}w_G (w_G + 1) = r, \\ \Sigma _G w_G ^2 = \bar w \\ \end{array}$$

hold, where ℐ hasv vertices, and$\bar w$ is the number of double cosets Γ^*σΓ^* (σ ∉ Γ^*). The second relation corresponds to the fact that the realizations associated with a given irreducible representationG form a cone of dimension 1/2w _G (w _G + 1), which forw _G ⩾ 2 has as base the convex hull of a projective space of dimensionw _G − 1 embedded in an ellipsoid of dimension 1/2w _G (w _G + 1) − 2. Comparison of the second and third relations leads to a curious connexion between the cone ℐ and the group Γ(ℐ), namely, that the following conditions are equivalent: (1)r =$\bar w$, (2) ℐ is polyhedral, (3)w _G ⩽ 1 for all irreducible orthogonal representationsG of Γ(ℐ), (4) σ^-1 ∈ Γ^*σΓ^* for each σ ∈ Γ(ℐ). The realization cones ℐ are described for various regular polytopes.