Editing Building (mathematics) (section)

==Connection with {{math|(''B'', ''N'')}} pairs==
If a group {{mvar|G}} acts simplicially on a building {{mvar|X}}, transitively on pairs {{math|(''C'',''A'')}} of chambers {{mvar|C}} and apartments {{mvar|A}} containing them, then the stabilisers of such a pair define a [[BN pair|{{math|(''B'', ''N'')}} pair]] or [[Tits system]]. In fact the pair of subgroups
:{{math|''B'' {{=}} ''G''<sub>''C''</sub>}} and {{mvar|''N'' {{=}} ''G''<sub>''A''</sub>}}
satisfies the axioms of a {{math|(''B'', ''N'')}} pair and the Weyl group can be identified with {{math|''N'' / ''N'' ∩ ''B''}}.

Conversely the building can be recovered from the {{math|(''B'', ''N'')}} pair, so that every {{math|(''B'', ''N'')}} pair canonically defines a building. In fact, using the terminology of {{math|(''B'', ''N'')}} pairs and calling any conjugate of {{mvar|B}} a [[Borel subgroup]] and any group containing a Borel subgroup a parabolic subgroup,
* the vertices of the building {{mvar|X}} correspond to maximal parabolic subgroups;
* {{math|''k'' + 1}} vertices form a {{mvar|k}}-simplex whenever the intersection of the corresponding maximal parabolic subgroups is also parabolic;
* apartments are conjugates under {{mvar|G}} of the simplicial subcomplex with vertices given by conjugates under {{mvar|N}} of maximal parabolics containing {{mvar|B}}.

The same building can often be described by different {{math|(''B'', ''N'')}} pairs. Moreover, not every building comes from a {{math|(''B'', ''N'')}} pair: this corresponds to the failure of classification results in low rank and dimension (see below).

The [[Solomon-Tits theorem]] is a result which states the homotopy type of a building of a group of Lie type is the same as that of a [[bouquet of spheres]].<ref>https://www.ams.org/journals/proc/1998-126-07/S0002-9939-98-04453-0/S0002-9939-98-04453-0.pdf</ref>