Abstract Regular Polytopes of Finite Irreducible Coxeter Groups

TL;DR

The paper investigates abstract regular polytopes arising from finite irreducible Coxeter groups using the string C-group framework. It provides a precise determination of the maximal and intermediate ranks for the $\text{D}_n$ series, showing $r_{ ext{max}}(\text{D}_n)=n-1$ when $n$ is even and $r_{ ext{D}_n}=n$ when $n$ is odd, and proves the existence of C-strings for all ranks $3\le r\le r_{ ext{max}}(\text{D}_n)$. It also supplies explicit rank-$3$ to rank-$r_{ ext{max}}$ constructions for $\text{D}_n$ via involutions in $\text{Sym}(2n)$ and employs rank-reduction techniques to obtain all intermediate ranks, along with a comprehensive census for exceptional groups where $r_{ ext{max}}(W)$ equals the Coxeter rank or specific small integers (5,6,7 for $\mathrm{E}_6$, $\mathrm{E}_7$, $\mathrm{E}_8$). The results extend the understanding of abstract regular polytopes associated with classical and exceptional Coxeter groups and provide computational data via Magma for the exceptional cases.

Abstract

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.

Theorems & Definitions (32)

• Theorem 1.1
• Theorem 1.2
• Theorem 1.3
• Lemma 2.1
• Lemma 2.2
• Lemma 2.3
• proof
• Theorem 2.4
• proof
• Theorem 2.5