Regular (flag-transitive) polytopes with few flags

Marston Conder (University of Auckland, New Zealand)

An abstract $n$-polytope is a partially-ordered set endowed with a rank function (plus a unique minimum and unique maximum element, of ranks $-1$ and $n$ respectively), such that all maximal chains have length $n+2$.  These are called flags.  Elements of ranks $0$, $1$, $2$ and $n-1$ are the vertices, edges, 2-faces and facets of the polytope. Also a certain geometrically-motivated `diamond' condition must hold, so that (for example) if a vertex is incident with a 2-face, then there are exactly two edges incident with both.  The polytope is called regular if its automorphism group has a single orbit on flags. In that case, the automorphism group is a smooth quotient of a Coxeter group $[p_1,p_2,\dots,p_n]$ where $p_1$ is the valency of each vertex, and so on. In this talk I will report on very recent work on finding for each $n$ the regular $n$-polytopes with the smallest numbers of flags, under the assumption that all $p_i > 2$. Somewhat surprisingly, for $n >3$ the smallest instances are not the regular $n$-simplices (of type $[3,3,\dots,3]$).