The Intersection Condition for Regular Polytopes
Marston Conder, University of Auckland, New Zealand
Location: Stevenson 1432
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.