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Journal of the London Mathematical Society Advance Access originally published online on December 6, 2007
Journal of the London Mathematical Society 2008 77(1):115-129; doi:10.1112/jlms/jdm093
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© 2007 London Mathematical Society

Constructions for chiral polytopes

Marston Conder

Department of Mathematics
University of Auckland
Private Bag 92019
Auckland
New Zealand

Isabel Hubard

Department of Mathematics and Statistics
York University
Toronto
Ontario
Canada M3J 1P3
isabel{at}yorku.ca

Tomaz Pisanski

IMFM
University of Ljubljana
Jadranska 19
1111 Ljubljana
Slovenia
Tomaz.Pisanski{at}fmf.uni-lj.si

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.

2000 Mathematics Subject Classification 52B15 (primary), 06A11, 20B25 (secondary).

The first author is supported in part by the N.Z. Marsden Fund, grant UOA 412. The second author is supported in part by CONACYT, Mexico. The third author is supported in part by the Ministry of Higher Education, Science and Technology of Slovenia, grants P1-0294, J1-6062, L1-7230.

Received March 28, 2007; published online December 6, 2007.


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