Regular polytopes and the intersection condition
Prof. Marston Conder, University of Auckland, New Zealand
An abstract polytope is a generalised form of a geometric polytope,
and may be viewed as a partially-ordered set (endowed with a rank
function) that satisfies certain properties motivated by the geometry.
A polytope is called regular if its automorphism group is transitive
(and hence sharply-transitive) on the set of all flags -- which are
the maximal chains in the poset. The automorphism group of a regular
polytope is a smooth quotient of a 'string' Coxeter group (with a
linear Dynkin diagram). Conversely, any finite smooth quotient of
such a group is the automorphism group of a regular polytope, provided
that it satisfies a condition known as the 'intersection condition'.
In this talk I will explain these things, and describe some recent
discoveries about the intersection condition, including its application
to find the smallest regular polytopes of any given rank.