On Tits Buildings of Type $A_n$

Abstract

Let $P$ and $P'$ be projective spaces having the same dimension $n$ assumed to be finite. Let $\Delta$ and $\Delta'$ be buildings of type $A_n$. We consider mappings of chambers of $\Delta$ into chambers of $\Delta'$ which send appartments to appartments. It is shown that shuch mappings are induced by strong embeddings of $P$ into $P'$ or into the dual space $P^{'*}$.