Abstract Regular Polytopes of Finite Irreducible Coxeter Groups

Here, for W the Coxeter group Dₙ 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 ≤ r ≤ n - 1, if n is even, and 3 ≤ r ≤ n, if n is odd. The possible ranks of abstract regular polytopes for the exceptional finite irreducible Coxeter groups are also determined.