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Monodromy Groups and Self-Invariance

Published online by Cambridge University Press:  20 November 2018

Isabel Hubard ,
Alen Orbanić  and
Asia Ivić Weiss
Isabel Hubard
Affiliation:
Department of Mathematics and Statistics, York University, Toronto, ON M3J 1P3 email: isabel@yorku.ca, weiss@yorku.ca
Alen Orbanić
Affiliation:
Faculty of Mathematics and Physics, University of Ljubljana, Jadranska 21, 1000 Ljubljana, Slovenia, EU email: alen.orbanic@fmf.uni-lj.si
Asia Ivić Weiss
Affiliation:
Department of Mathematics and Statistics, York University, Toronto, ON M3J 1P3 email: isabel@yorku.ca, weiss@yorku.ca
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Abstract

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For every polytope $\mathcal{P}$ there is the universal regular polytope of the same rank as $\mathcal{P}$ corresponding to the Coxeter group $\mathcal{C}\,=\,\left[ \infty ,\,.\,.\,.\,,\,\infty \right]$. For a given automorphism $d$ of $\mathcal{C}$, using monodromy groups, we construct a combinatorial structure ${{P}^{d}}$. When ${{P}^{d}}$ is a polytope isomorphic to $\mathcal{P}$ we say that $\mathcal{P}$ is self-invariant with respect to $d$, or $d$-invariant. We develop algebraic tools for investigating these operations on polytopes, and in particular give a criterion on the existence of a $d$-automorphism of a given order. As an application, we analyze properties of self-dual edge-transitive polyhedra and polyhedra with two flag-orbits. We investigate properties of medials of such polyhedra. Furthermore, we give an example of a self-dual equivelar polyhedron which contains no polarity (duality of order 2). We also extend the concept of Petrie dual to higher dimensions, and we show how it can be dealt with using self-invariance.

Keywords

51M2005C2505C1005C3052B70mapsabstract polytopesself-dualitymonodromy groupsmedials of polyhedra
Type
Research Article
Information
Canadian Journal of Mathematics , Volume 61 , Issue 6 , 01 December 2009 , pp. 1300 - 1324
Copyright
Copyright © Canadian Mathematical Society 2009

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