On opposition in spherical buildings and twin buildings
The notion of twin buildings, as introduced by Ronan and Tits [see \textit{J. Tits}, Lond. Math. Soc. Lect. Note Ser. 165, 249-286 (1992; Zbl 0851.22023)], is a generalization of spherical buildings. One considers a pair of buildings \((\Delta_+,\Delta_-)\) of the same type and a codistance function \(\delta^*\) defined on pairs of chambers, one in \(\Delta_+\) and the other in \(\Delta_-\), with values in the Weyl group \(W\). Opposite chambers are chambers of codistance 1 (the identity in \(W\)). The role of apartments in spherical buildings is taken by twin apartments in twin buildings; these are pairs of apartments \(\Sigma_+\) in \(\Delta_+\) and \(\Sigma_-\) in \(\Delta_-\) that have the property that each chamber of one is opposite a unique chamber in the other. Based on the notion of opposition of chambers the authors characterize twin apartments and adjacency of chambers in twin buildings in the paper under review. They show that a non-empty set \({\mathcal M}\) of chambers of a thick twin building \(\Delta\) is the chamber set of a twin apartment if and only if for every chamber \(C\) in \({\mathcal M}\) there is a unique chamber in \({\mathcal M}\) that is opposite \(C\), and for every chamber \(C\) of \(\Delta\) not belonging to \({\mathcal M}\) the number of chambers in \({\mathcal M}\) that are opposite to \(C\) is even. Furthermore, two distinct chambers \(C\) and \(D\) of \(\Delta\) are adjacent if and only if there exists a chamber \(E\) in \(\Delta\) such that no chamber of \(\Delta\) is opposite exactly one of \(\{C,D,E\}\). The authors further study maps defined on the set of chambers of a thick 2-spherical twin building (that is, every rank 2 residue is of spherical type) that preserve opposition and non-opposition. If such a map into another thick twin building is surjective, then it can be extended to an isomorphism of twin buildings. In particular, Weyl distances and codistances are uniquely determined by the opposition relation on the set of chambers of a thick 2-spherical twin building. Since every spherical building can be twinned to itself to obtain a twin building in such a way that the notions of opposition of chambers, apartments and adjacency of chambers in the spherical and the associated twin building agree, the author's results have interpretations for spherical buildings. The results obtained in this way for spherical buildings are new.
scientific article; zbMATH DE number 1491016