Enumerating A3(2) Blueprints, and an Application

1. Definitions 2. Enumerating Blueprints 3. Applications References We present an enumeration scheme for all blueprints of an A2(p) building (with p prime). We then provide a computer based proof that the e An(Q 2) buildings do not conform to a blueprint for n 3. J. Tits ([1955b; 1955a; 1956]; see the introduction of [Tits 1974] for a historical discussion) introduced buildings in an attempt to provide a systematic procedure for producing geometries associated to semisimple Lie groups. With this procedure, Tits found geometries associated to the exceptional groups E6, E7, and E8. One of the early hopes for buildings was to provide a way to combinatorially construct the nite groups of Lie type, and in [Ronan and Tits 1987], this was accomplished using blueprints. Ronan and Tits, however, were unable to nd blueprints constructing general padic a ne buildings. At the end of their paper though, they suggested a generalization of a blueprint which might be used. Abramson [ 1998] presented a generalization of a blueprint like that suggested by Ronan and Tits. He used this to construct the p-adic buildings. The question of whether in general p-adic buildings conform to blueprints was left unanswered. Currently there is only one known case of a padic building with a blueprint. Ronan ([1984]; see [Kantor 1986] for a complete discussion) presented four buildings of type e A2 associated to the Frobenius group of order 21. P. K ohler, T. Meixner, and M. Wester proved that one of these corresponded to the classical e A2 building associated to the eld F 2((t)) [Kohler et al. 1984] and a second corresponded to the 2-adic building of type e A2 over the