Abstract Regular Polytopes of Finite Irreducible Coxeter Groups

Here, for $W$ the Coxeter group $\mathrm{D}_n$ where $n > 4$, it is proved that the maximal rank of an abstract regular polytope for $W$ is $n - 1$ if $n$ is even and $n$ if $n$ is odd. Further it is shown that $W$ has abstract regular polytopes of rank $r$ for all $r$ such that $3 \leq r \leq n - 1$, if $n$ is even, and $3 \leq r \leq n$, if $n$ is odd. The possible ranks of abstract regular polytopes for the exceptional finite irreducible Coxeter groups are also determined.

💡 Research Summary

The paper investigates abstract regular polytopes associated with finite irreducible Coxeter groups by means of string C‑groups, i.e., groups generated by involutions satisfying the commuting (|i‑j|≥2) and intersection properties. The authors focus primarily on the Dₙ family (n>4) and determine both the maximal possible rank rₘₐₓ of a string C‑group for Dₙ and the existence of such groups for every intermediate rank.

First, Dₙ is realized as a subgroup of the symmetric group Sym(2n) using explicit transpositions β₀,…,βₙ₋₁ (Lemma 2.1). This embedding allows the authors to apply known results on permutation groups: Whiston’s theorem (the size of an independent generating set in Sym(m) is at most m‑1) and a bound for the rank of a transitive string C‑group (Theorem 2.5), which together give an upper bound for rₘₐₓ. By a careful analysis of independent generating subsets T⊂S and the intersection condition, they show that when n is even, a rank‑n C‑group cannot exist, so rₘₐₓ=n‑1; when n is odd, a rank‑n C‑group does exist, giving rₘₐₓ=n.

To prove existence, the authors construct a family of involutions t₁,…,tₙ in Sym(2n). Lemma 3.2 computes the orders of products tᵢtⱼ, establishing the required commuting relations. Lemma 3.3 proves by induction that ⟨t₁,…,tₖ⟩ is a string C‑group for every k≥3. Lemma 3.4 shows that for odd n the whole set ⟨t₁,…,tₙ⟩ is isomorphic to Dₙ, and Lemma 3.5 confirms that it has rank n with Schlӓfli type {4,3ⁿ⁻²}. Thus the maximal rank is attained.

For lower ranks, the paper invokes the rank‑reduction theorem of Brooksbank and Leemans (Theorem 2.8). This theorem permits the systematic removal of generators while preserving the string C‑group structure, provided certain oddness conditions on the Schlӓfli symbols are met. Applying this repeatedly yields string C‑groups of every rank r with 3≤r≤rₘₐₓ, completing the existence part of the main theorem.

The study also treats the exceptional finite irreducible Coxeter groups. For I₂(m), H₃, H₄, and F₄, the maximal rank coincides with the Coxeter rank. For the three exceptional E‑type groups, the maximal ranks are 5, 6, 7 respectively. These results align with earlier computational atlases and confirm conjectures about the limitations imposed by the group’s internal structure.

Overall, the paper combines permutation‑group techniques, independent set bounds, explicit involution constructions, and rank‑reduction methods to give a complete classification of possible ranks of abstract regular polytopes for all finite irreducible Coxeter groups. The results deepen the connection between Coxeter theory and the combinatorial geometry of regular polytopes, and they provide a robust framework that can be extended to other families of groups or to the study of geometric realizations of the corresponding polytopes.