Highest rank of a polytope for $A_n$

TL;DR

This paper determines the maximum rank of string C-groups derived from alternating groups $Alt_n$, providing exact values for small n and a general formula for larger n, thus resolving a conjecture from 2012.

Contribution

The paper establishes the exact highest rank of string C-groups for all alternating groups, confirming a conjecture and extending understanding of polytope symmetries.

Findings

Highest rank for small n: 0 for n=3,4,6,7,8; 3 for n=5; 4 for n=9; 5 for n=10; 6 for n=11.

For n ≥ 12, the highest rank is floor((n-1)/2).

The conjecture from 2012 is fully proved.

Abstract

We prove that the highest rank of a string C-group constructed from an alternating group $Alt_n$ 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 $n\geq 12$. This solves a conjecture made by the last three authors in 2012.