Highest rank of a polytope for $A_n$

Abstract

We prove that the highest rank of a string C-group constructed from an alternating group $Al{t}_n$ is 0 if $n=3,4,6,7,8$; 3 if $n=5$; 4 if $n=9$; 5 if $n=10$; 6 if $n=11$; and $\lfloor \frac{n-1}{2}\rfloor$ if $n\ge 12$. This solves a conjecture made by the last three authors in 2012.