Weak buildings of spherical type (Q1110805)

Thick buildings of spherical type of rank \(\geq 3\) have been classified by \textit{J. Tits} [Buildings of spherical type and finite BN-pairs (1974; Zbl 0295.20047)]; these are the buildings associated with groups with BN- pairs. Thin buildings are precisely the Coxeter complexes. In this note the author deals with weak (i.e. non-thick) buildings of spherical type and presents a construction of a certain class of these buildings via associated point-line geometries. To every weak building \(\Delta\) of rank n and each \(i\in I=\{1,...,n\}\) there is an associated point-line geometry, the standard i-geometry (shadow geometry) \(\Delta_ i\) of \(\Delta\) ; points of \(\Delta_ i\) are the vertices of \(\Delta\) of type i and lines of \(\Delta_ i\) are the sets of all those points of \(\Delta_ i\) that can be joined to a given simplex of type \(I\setminus \{i\}\). Starting from the Coxeter complex \(\Sigma\) of a finite Coxeter group W with generators \(r_ 1,...,r_ n\), two constructions of point-line geometries \(\Delta_ i(\Sigma_ i,\Gamma)\) and \(\Delta_ i(\Sigma_ i,\Gamma,\Lambda)\) are given for every \(i\in I=\{1,...,n\}\) involving the standard i-geometry \(\Sigma_ i\) of \(\Sigma\), a rank n-1 building \(\Gamma\) whose appartements are all isomorphic to the smaller Coxeter complex \(\Sigma (W\setminus \{r_ i\})\) and an indexing set \(\Lambda\) of at least two elements. The construction of \(\Delta_ i(\Sigma_ i,\Gamma,\Lambda)\) is feasible in special cases only; then it is a generalization of \(\Delta_ i(\Sigma_ i,\Gamma)\) which is obtained if \(\Lambda\) contains just two elements. In the constuctions the standard i-graph \(\Sigma_ i\) is decomposed in terms of Coxeter graphs of a single lower rank Coxeter complex which then is substituted by a building of the same type to obtain the point-line geometry. For these resulting geometries (for fixed i) the author shows that they are isomorphic to the standard i-geometry of a weak building \(\Delta\) of the same type and rank as \(\Gamma\). Both constructions result in point-line geometries for which there is a point that is only on thin lines (i.e. on lines with just two points). In general, for any weak building \(\Delta\) there is an index i such that the standard i-geometry \(\Delta_ i\) has this property. However, as it is shown by a simple example in the note, the given constructions do not constitute for all point-line geometries of weak buildings. Finally the author discusses to what extent weak buildings of spherical type have standard i-geometries that are isomorphic to the constructed point-line geometries and, in certain low rank cases, she obtaines a full description. A different approach to weak buildings of spherical type has been adopted by \textit{R. Scharlau} [Geom. Dedicata 24, 77-84 (1987; Zbl 0644.51009)]; here to every building \(\Delta\) there is canonically associated a thick building \({\bar \Delta}\) whose Weyl group W(\({\bar \Delta}\)) can be considered as a reflection subgroup of the Weyl group W(\(\Delta)\) of \(\Delta\). From the embedding of W(\({\bar \Delta}\)) in W(\(\Delta)\), the original building \(\Delta\) may be reconstructed from \({\bar \Delta}\).