The smallest regular hypertopes of various kinds

Marston Conder
University of Auckland, New Zealand

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Minisymposium: POLYTOPES AND GRAPHS

Content: A {\em hypertope\/} is a generalisation of an abstract polytope, just as hypermaps generalise maps. Formally, a hypertope is a residually connected thin incidence geometry, with elements categorised according to their type. The theory of hypertopes has been developed recently by Fernandes, Leemans and Weiss. The rank of a hypertope ${\cal H}$ is the number of distinct types. A {\em flag\/} is a set of pairwise incident elements, or equivalently, a clique in the incidence graph, and if this contains one element of each type, then it is called a {\em chamber\/} of ${\cal H}$. For the purposes of this talk, the automorphism group ${\rm Aut}(\cal H)$ of a hypertope ${\cal H}$ is the group of all incidence- and type-preserving bijections on ${\cal H}$, and then ${\cal H}$ is called {\em regular\/} if ${\rm Aut}(\cal H)$ is transitive on the set of all chambers of ${\cal H}$. % When this happens, ${\rm Aut}(\cal H)$ is a smooth quotient of some Coxeter group. In the special case where $\cal H$ is a polytope, the underlying graph of the Coxeter/Dynkin diagram for the latter group is a path, but for general hypertopes, it can be any connected graph. In this talk, I will describe some recent work with Dimitri Leemans and Daniel Pellicer in which we found the smallest regular hypertopes with a given graph underlying its Coxeter/Dynkin diagram. These are already known for all finite paths (namely the smallest regular polytopes of each rank), and so we concentrated on the following graphs: cycles $C_n$ for $n \ge 3$, star graphs $S_n$ for $n \ge 3$, and complete graphs $K_n$ for $n \ge 3$, as well as the underlying graphs of the Coxeter/Dynkin diagrams of types $D_n$ (for $n \ge 4$), ${\tilde D}_n$ (for $n \ge 5$), $E_n$ (for $n \ge 6$), and ${\tilde E}_6$, ${\tilde E}_7$ and ${\tilde E}_8$. Some of what we discovered is a little surprising.

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