October 21, 2025Open Access

Abstract Regular Polytopes of Finite Irreducible Coxeter Groups

Key Points

The maximal rank of abstract regular polytopes for D_n is n - 1 if n is even, and n if n is odd.
Polytopes of rank r exist for all r from 3 to n - 1 when n is even, and from 3 to n when n is odd.
Ranks of abstract regular polytopes for exceptional finite irreducible Coxeter groups have also been established.
The work provides insights into the structural properties of polytopes associated with Coxeter groups.

Abstract

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.

Abstract Regular Polytopes of Finite Irreducible Coxeter Groups

Key Points

Abstract

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