20 Group theory and generalizations -- 20G Linear algebraic groups and related topics 20G25 Linear algebraic groups over local fields and their integers
51 Geometry -- 51E Finite geometry and special incidence structures 51E24 Buildings and the geometry of diagrams
M. A. Ronan\ and J. Tits\ [Math. Ann. {\bf 278} (1987), no.~1-4, 291--306; MR0909229 (89e:51005)] introduced ``blueprints'', i.e. triples made of a Coxeter group $(W,S)$, a valency function $q$ on $S$ with values in $\bold N$ (or more generally in the ``set'' of cardinals) and some rank two equivalences. To such a blueprint is associated a chamber complex of type $W$ and the blueprint is said to be realizable if this complex is a building. When $W$ is finite Ronan and Tits used these blueprints to build abstractly the buildings of the finite groups of Lie type. If $q$ is constant equal to a prime $p$, if the rank two equivalences are ``natural'' and if $W$ is the affine Weyl group of type $\tilde X_n$ (simply laced with $n\geq 3$), they proved that the associated building is the Bruhat-Tits building of type $X_n$ over $\bold F_q((t))$. Now the problem is whether the Bruhat-Tits buildings over $\bold Q_p$ are associated to blueprints. If one slightly modifies the definition of blueprints the answer is ``yes'' [M. A. Abramson, Geom. Dedicata {\bf 76} (1999), no.~2, 127--154 1703209 ]. \par With the original definition of blueprints, the authors prove that the answer is ``no'' for $p=2$, $n\geq 3$ and $\tilde X_n$ simply laced: in this case all blueprint equivalences are ``natural''. For $p$ a prime the authors prove first that all blueprints associated to the same building of type $A_2$ are obtained from each other by suitable relabelings. Then a computer-based proof implies that all blueprints associated to the Tits building of type $A_3$ over $\bold F_2$ have ``natural'' equivalences. The above result follows.