Abstract Regular Polytopes of Finite Irreducible Coxeter Groups

TL;DR

This paper investigates the ranks of abstract regular polytopes associated with finite irreducible Coxeter groups, establishing maximum ranks and existence conditions for various ranks across different group types.

Contribution

It determines the maximum possible ranks of abstract regular polytopes for Coxeter groups of type D and classifies possible ranks for exceptional finite irreducible Coxeter groups.

Findings

Maximum rank for type D groups is n-1 if n even, n if n odd.

Existence of polytopes of all ranks within certain ranges depending on n.

Ranks for exceptional Coxeter groups are explicitly classified.

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.