yes, you are right,
algebraic conjugacy here indeed provides
different representations of the
same abstract polytopes.
In fact, you could freely vary in a Dynkin diagram
- 4 vs. 4/3
- 5 vs. 5/3
- 5/2 vs. 5/4
while the incidence matrix will not change at all.
But the vertex set surely would change thereby and hence those representations do belong to different armies.
--- rk