Building (mathematics) - Connection With BN Pairs

Connection With BN Pairs

If a group G acts simplicially on a building X, transitively on pairs of chambers C and apartments A containing them, then the stabilisers of such a pair define a BN pair or Tits system. In fact the pair of subgroups

B = G_C and N = G_A

satisfies the axioms of a BN pair and the Weyl group can identified with N / N B. Conversely the building can be recovered from the BN pair, so that every BN pair canonically defines a building. In fact, using the terminology of BN pairs and calling any conjugate of B a Borel subgroup and any group containing a Borel subgroup a parabolic subgroup,

• the vertices of the building X correspond to maximal parabolic subgroups;
• k + 1 vertices form a k-simplex whenever the intersection of the corresponding maximal parabolic subgroups is also parabolic;
• apartments are conjugates under G of the simplicial subcomplex with vertices given by conjugates under N of maximal parabolics containing B.

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