ISSN:
1420-8903
Keywords:
Primary 51M20
Source:
Springer Online Journal Archives 1860-2000
Topics:
Mathematics
Notes:
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.
Type of Medium:
Electronic Resource
URL:
http://dx.doi.org/10.1007/BF01837943
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