New PDF release: Buildings (Cambridge University Part III)
By Brian Lehmann
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Additional info for Buildings (Cambridge University Part III)
It turns out that every spherical building yields a foundation. Furthermore, these foundations each yield a unique building provided that they satisfy some conditions on their restriction to spherical buildings of rank 3. There are only a few such restrictions to consider, and analyzing these leads to a full classification. 29 Another important case is that of affine buildings — those whose Coxeter complexes correspond to affine reflection groups. For such buildings, each apartment will correspond to a tiling of a Euclidean vector space.
1989)  Brown, Kenneth Buildings, Springer-Verlag New York Inc.
In fact, by including a labeling scheme on the chambers, it is possible in many cases to collate all of the important information about the building into a foundation consisting of these J-residues and the labeling information. Foundations allow us to reconstruct the building by gluing together the J-residues according to an equivalency determined by the labeling. It turns out that every spherical building yields a foundation. Furthermore, these foundations each yield a unique building provided that they satisfy some conditions on their restriction to spherical buildings of rank 3.
Buildings (Cambridge University Part III) by Brian Lehmann