BN pair

In mathematics, a (B, N) pair is a structure on groups of Lie type that allows one to give uniform proofs of many results, instead of giving a large number of case-by-case proofs. Roughly speaking, it shows that all such groups are similar to the general linear group over a field. They were invented by the mathematician Jacques Tits, and are also sometimes known as Tits systems.

A (B, N) pair is a pair of subgroups B and N of a group G such that the following axioms hold:

The idea of this definition is that B is an analogue of the upper triangular matrices of the general linear group GL_n(K), H is an analogue of the diagonal matrices, and N is an analogue of the normalizer of H.

The subgroup B is sometimes called the Borel subgroup, H is sometimes called the Cartan subgroup, and W is called the Weyl group. The pair (W,S) is a Coxeter system.

The number of generators is called the rank.

The map taking w to BwB is an isomorphism from the set of elements of W to the set of double cosets of B; this is the Bruhat decomposition G = BWB.

If T is a subset of S then let W(T) be the subgroup of W generated by T: we define and G(T) = BW(T)B to be the standard parabolic subgroup for T. The subgroups of G containing conjugates of B are the parabolic subgroups; conjugates of B are called Borel subgroups (or minimal parabolic subgroups). These are precisely the standard parabolic subgroups.

BN-pairs can be used to prove that many groups of Lie type are simple modulo their centers. More precisely, if G has a BN-pair such that B is a solvable group, the intersection of all conjugates of B is trivial, and the set of generators of W cannot be decomposed into two non-empty commuting sets, then G is simple whenever it is a perfect group. In practice all of these conditions except for G being perfect are easy to check. Checking that G is perfect needs some slightly messy calculations (and in fact there are a few small groups of Lie type which are not perfect). But showing that a group is perfect is usually far easier than showing it is simple.

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