Abstract Regular Polytopes of Finite Irreducible Coxeter Groups

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 $3\le r\le n-1$, if $n$ is even, and $3\le r\le n$, if $n$ is odd. The possible ranks of abstract regular polytopes for the exceptional finite irreducible Coxeter groups are also determined.