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.
@article{10_37236_13763,
author = {Malcolm Hoong Wai Chen and Peter Rowley},
title = {Abstract regular polytopes of finite irreducible {Coxeter} groups},
journal = {The electronic journal of combinatorics},
year = {2025},
volume = {32},
number = {4},
doi = {10.37236/13763},
zbl = {8120104},
url = {http://geodesic.mathdoc.fr/articles/10.37236/13763/}
}
TY - JOUR
AU - Malcolm Hoong Wai Chen
AU - Peter Rowley
TI - Abstract regular polytopes of finite irreducible Coxeter groups
JO - The electronic journal of combinatorics
PY - 2025
VL - 32
IS - 4
UR - http://geodesic.mathdoc.fr/articles/10.37236/13763/
DO - 10.37236/13763
ID - 10_37236_13763
ER -
%0 Journal Article
%A Malcolm Hoong Wai Chen
%A Peter Rowley
%T Abstract regular polytopes of finite irreducible Coxeter groups
%J The electronic journal of combinatorics
%D 2025
%V 32
%N 4
%U http://geodesic.mathdoc.fr/articles/10.37236/13763/
%R 10.37236/13763
%F 10_37236_13763
Malcolm Hoong Wai Chen; Peter Rowley. Abstract regular polytopes of finite irreducible Coxeter groups. The electronic journal of combinatorics, Tome 32 (2025) no. 4. doi: 10.37236/13763
