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Barry Monson - Realizations of regular abstract polytopes



BARRY MONSON, Department of Mathematics and Statistics, University of New Brunswick, Fredericton, New Brunswick  E3B 5A3, Canada
Realizations of regular abstract polytopes


A regular abstract polytope ${\cal P}$ is a poset having the `essential' structural features of the face lattice of a regular convex polytope, including transitivity of $\textrm{Aut}({\cal P})$ on flags. Other examples include regular tessellations and honeycombs, star-polyhedra and many less familiar beasts. Since ${\cal P}$ need not actually be a lattice or have a particularly nice geometric realization, it is interesting to describe all realizations for ${\cal P}$. The basic theory of realizations was developed by McMullen (1989), who used geometric methods to describe a crucial family of real representations for $\textrm{Aut}({\cal P})$.

After an explicit, but brief, description of these objects, I will discuss work with Asia Ivic Weiss, in which we construct families of regular polytopes parametrized by certain rings of algebraic integers. Typically, our understanding of the realization cones for these polytopes is very limited.


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Next: Konstantin Rybnikov - On Up: Discrete Geometry / Géométrie Previous: Wodzimierz Kuperburg - Covering
 

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