# The smallest regular hypertopes of various kinds

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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 ${\cal H}$.
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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.