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Constructions for chiral polytopes
Author(s) -
Conder Marston,
Hubard Isabel,
Pisanski Tomaž
Publication year - 2008
Publication title -
journal of the london mathematical society
Language(s) - English
Resource type - Journals
SCImago Journal Rank - 1.441
H-Index - 62
eISSN - 1469-7750
pISSN - 0024-6107
DOI - 10.1112/jlms/jdm093
Subject(s) - coxeter group , polytope , combinatorics , rank (graph theory) , mathematics , dual (grammatical number) , group (periodic table) , automorphism , physics , art , literature , quantum mechanics
An abstract polytope of rank n is said to be chiral if its automorphism group has two orbits on flags, with adjacent flags lying in different orbits. In this paper, we describe a method for constructing finite chiral n ‐polytopes, by seeking particular normal subgroups of the orientation‐preserving subgroup of an n ‐generator Coxeter group (having the property that the subgroup is not normalized by any reflection and is therefore not normal in the full Coxeter group). This technique is used to identify the smallest examples of chiral 3‐ and 4‐polytopes, in both the self‐dual and non‐self‐dual cases, and then to give the first known examples of finite chiral 5‐polytopes, again in both the self‐dual and non‐self‐dual cases.