A characterization of twin buildings by twin apartments
Spherical buildings are characterised by their apartments and opposite chambers. In non-spherical buildings, however, one no longer has pairs of opposite chambers. To remedy this situation twin buildings, where chambers in one building can be opposite chambers of another, were introduced. The authors start from the definition of a twin building \((\Delta_+, \Delta_-, \delta^*)\) as given by \textit{J. Tits} [Lond. Math. Soc. Lect. Note Ser. 165, 249-286 (1992; Zbl 0851.22023)] consisting of two buildings \(\Delta_+\) and \(\Delta_-\) of the same type \(M\) and a \(W\)-codistance function \(\delta^*\) defined on \({\mathcal C}_+\times{\mathcal C}_-\cup{\mathcal C}_-\times{\mathcal C}_+\) with values in the Weyl group \(W\). Opposite chambers are chambers of codistance 1. The authors develop a new system of axioms for twin buildings (without the use of a \(W\)-codistance function) more along the lines of Tits' original definition of spherical buildings where the role of spherical apartments is taken by twin apartments. Let \(\Delta=(\Delta_+, \Delta_-,{\mathcal A},\text{op})\) where \(\Delta_+\) and \(\Delta_-\) are two buildings of the same type with respective sets of chambers \({\mathcal C}_+\) and \({\mathcal C}_-\), the set \({\mathcal A}\) of twin apartments is a subset of \(\{(\Sigma_+,\Sigma_-) | \Sigma_\varepsilon\) is an apartment of \(\Delta_\varepsilon, \varepsilon\in\{+,-\}\}\) and \(\text{op}\subseteq {\mathcal C}_+\times{\mathcal C}-\cup{\mathcal C}_-\times{\mathcal C}_+\) is a symmetric opposition relation such that the following four conditions are satisfied: (TA1) For every \(\Sigma=(\Sigma_+,\Sigma_-)\in{\mathcal A}\), the opposition relation induces an isomorphism op\(_\Sigma:\Sigma_+\to\Sigma_- \) between Coxeter complexes. (TA2) For all \(c_+\in{\mathcal C}_+, d_-\in{\mathcal C}_-\), there exists a \(\Sigma\in{\mathcal A}\) such that \((c_+,d_-)\in\Sigma\). (TA3) For all \(\Sigma, \widetilde\Sigma\in{\mathcal A}\) and all \(a=(a_+,a_-)\in\Sigma\cap\widetilde\Sigma\), there exists an isomorphism \(\alpha:\Sigma\to\widetilde\Sigma\) satisfying \(\alpha(a)=a\), that is, \(\alpha =(\alpha_+,\alpha_-)\) is a pair of isomorphisms of Coxeter complexes satisfying \(\text{op}_{\widetilde\Sigma}\circ\alpha_+=\alpha_-\circ\text{op} _\Sigma\). (TA4) For all \(\Sigma, \widetilde\Sigma\in{\mathcal A}\) such that \(\Sigma\cap\widetilde\Sigma\) contains a chamber, \(\Sigma\cap\widetilde\Sigma\) is coconvex in \(\Sigma\) and in \(\widetilde\Sigma\). The axioms (TA1)-(TA3) are sufficient to construct a `codistance-like' function \(\delta^*\) such that \((\Sigma,\delta^*| _\Sigma)\) is a twin building for each \(\Sigma\in {\mathcal A}\) so that coconvexity is already defined in \(\Sigma\). The main result of the authors then says that \(\Delta=(\Delta_+, \Delta_-,{\mathcal A},\text{ op})\) satisfies (TA1)-(TA4) if and only if there exists a codistance function \(\delta^*\) such that \((\Delta_+, \Delta_-, \delta^*)\) is a twin building and \({\mathcal A}\) is the collection of opposite chambers. They further discuss various variations of the above system of axioms and give a counter-example that shows that axioms (TA1)--(TA3) do not suffice to characterise twin buildings.
scientific article; zbMATH DE number 1240983