Abstract view
Monodromy Groups and SelfInvariance


Published:20091201
Printed: Dec 2009
Isabel Hubard
Alen Orbani\'c
Asia Ivi\'c Weiss
Abstract
For every polytope $\mathcal{P}$ there is the universal regular
polytope of the same rank as $\mathcal{P}$ corresponding to the
Coxeter group $\mathcal{C} =[\infty, \dots, \infty]$. For a given
automorphism $d$ of $\mathcal{C}$, using monodromy groups, we
construct a combinatorial structure $\mathcal{P}^d$. When
$\mathcal{P}^d$ is a polytope isomorphic to $\mathcal{P}$ we say that
$\mathcal{P}$ is selfinvariant 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 $d$\nobreakdashauto\morphism of a given order. As an application,
we analyze properties of selfdual edgetransitive polyhedra and
polyhedra with two flagorbits. We investigate properties of medials
of such polyhedra. Furthermore, we give an example of a selfdual
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 selfinvariance.
MSC Classifications: 
51M20, 05C25, 05C10, 05C30, 52B70 show english descriptions
Polyhedra and polytopes; regular figures, division of spaces [See also 51F15] Graphs and abstract algebra (groups, rings, fields, etc.) [See also 20F65] Planar graphs; geometric and topological aspects of graph theory [See also 57M15, 57M25] Enumeration in graph theory Polyhedral manifolds
51M20  Polyhedra and polytopes; regular figures, division of spaces [See also 51F15] 05C25  Graphs and abstract algebra (groups, rings, fields, etc.) [See also 20F65] 05C10  Planar graphs; geometric and topological aspects of graph theory [See also 57M15, 57M25] 05C30  Enumeration in graph theory 52B70  Polyhedral manifolds
