Problems on polytopes, their groups, and realizations
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OBLEMS ON POLYTOPES, THEIR GROUPS, AND REALIZATIONS ´ Weiss2,∗∗ (Toronto) Egon Schulte1,∗ (Boston) and Asia Ivic 1
Northeastern University, Boston, MA 02115, USA, E-mail: [email protected]
2
York University, Toronto, ON, M3J 1P3, Canada E-mail: [email protected]
(Received: May 17, 2005; Accepted: May 23, 2006)
Abstract The paper gives a collection of open problems on abstract polytopes that were either presented at the Polytopes Day in Calgary or motivated by discussions at the preceding Workshop on Convex and Abstract Polytopes at the Banff International Research Station in May 2005.
1. Introduction The rapid development of polytope theory in the past thirty years has resulted in a rich theory featuring an attractive interplay of several mathematical disciplines. The breadth of the talks at the Workshop on Convex and Abstract Polytopes and the subsequent Polytopes Day in Calgary that we organized jointly with Ted Bisztriczky at the Banff International Research Station (BIRS) on May 19–21, 2005 and the University of Calgary on May 22, 2005, respectively, gave evidence that polytope theory is very much alive and is the unifying theme of a lot of research activity. The Workshop provided a much desired opportunity to share recent developments and Mathematics subject classification number: 51M20, 52B15, 20F55. Key words and phrases: polytopes, polyhedra, symmetry, regular, chiral, reflection groups, C-groups. ∗ Supported by NSA-grant H98230-05-1-0027. ∗∗ Supported by NSERC of Canada Grant #8857. 0031-5303/2006/$20.00 c Akad´ emiai Kiad´ o, Budapest
Akad´ emiai Kiad´ o, Budapest Springer, Dordrecht
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e. schulte and a. i. weiss
emerging directions on geometric, combinatorial, and abstract aspects of polytope theory. It is noteworthy that the last major meeting on convex and abstract polytopes was the NATO Advanced Study Institute on Polytopes – Abstract, Convex and Computational in 1993 in Scarborough, Ontario (see [3]). For abstract polytopes, the invited lectures and talks focused on polytopes with various degrees of combinatorial or geometric symmetry (regular, chiral, or equivelar polytopes, and their geometric realization theory), as well as the structure of their symmetry groups or automorphism groups (reflection groups, Coxeter groups, and C-groups, and their representation theory). The present paper surveys open research problems on abstract polytopes that were presented at the Workshop (primarily at the problem session). A significant number of these problems have been addressed in detail elsewhere in the literature, notably in [43]. For the remaining problems we provide some background information when available, but due to space limitations we cannot give a comprehensive account in all cases. In Section 2 we review basic notions and concepts, and then in the subsequent sections explore some of the most important problems in the field. It is natural to group these problems under the headings and subheadings provided in Sections 3 to 6. There are, however, a number of interesting research problems tha
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