Pure Maths Colloquium: Peter Cameron

An abstract polytope is a combinatorial object abstracted from the notions of polygons and polyhedra; we keep data about the incidence of objects (vertices, edges, faces, … ) but discard all metric data. Thus it has objects of all possible dimensions from \(0\) to \(r − 1\), where \(r\) is the rank.

A polytope is regular if its automorphism group acts transitively (and hence regularly) on the set of maximal flags (sets of mutually incident objects). For example, the \(r\)-dimensional simplex can be represented by taking the \(i\)-dimensional objects to be the \((i + 1)\)- element sets of an \((r+1)\)-set; its automorphism group is the symmetric group of degree \(n = r + 1\). It can be shown that a polytope with automorphism group \(S_n\) has rank at most \(n − 1\), with equality if and only if it is the simplex.

After a sequence of results over the last decade, Maria Elisa Fernandes (Averio), Dimitri Leemans (Brussels) and I have proved:

Theorem: For any positive integer \(k\), there is a positive integer \(c_k\) such that the number of regular polytopes with automorphism group \(S_n\) and rank \(n − k\) (up to isomorphism and duality) is \(c_k\) (independent of \(n\)) for \(n\ge 2k+3\).

The sequence \((c_k)\) begins \(1, 1, 7, 9, 35, 48, \ldots\). (No further terms are currently known.)

I will talk mostly about the background and history of this problem, with only a very brief sketch of the proof; the paper is over 40 pages long.