(B, N) 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 introduced by the mathematician Jacques Tits, and are also sometimes known as Tits systems.

Definition

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

G is generated by B and N.
The intersection, H, of B and N is a normal subgroup of N.
The group W = N/H is generated by a set S of elements w_i of order 2, for i in some non-empty set I.
If w_i is an element of S and w is any element of W, then w_iBw is contained in the union of Bw_iwB and BwB.
No generator w_i normalizes B.

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.

Examples

Suppose that G is any doubly transitive permutation group on a set X with more than 2 elements. We let B be the subgroup of G fixing a point x, and we let N be the subgroup fixing or exchanging 2 points x and y. The subgroup H is then the set of elements fixing both x and y, and W has order 2 and its nontrivial element is represented by anything exchanging x and y.
Conversely, if G has a (B, N) pair of rank 1, then the action of G on the cosets of B is doubly transitive. So BN pairs of rank 1 are more or less the same as doubly transitive actions on sets with more than 2 elements.
Suppose that G is the general linear group GL_n(K) over a field K. We take B to be the upper triangular matrices, H to be the diagonal matrices, and N to be the monomial matrices, i.e. matrices with exactly one non-zero element in each row and column. There are n − 1 generators w_i, represented by the matrices obtained by swapping two adjacent rows of a diagonal matrix.
More generally, any group of Lie type has the structure of a BN-pair.
A reductive algebraic group over a local field has a BN-pair where B is an Iwahori subgroup.

Properties of groups with a BN pair

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.

Applications

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.

References

Bourbaki, Nicolas (2002). Lie Groups and Lie Algebras: Chapters 4–6. Elements of Mathematics. Springer. ISBN 3-540-42650-7. Zbl 0983.17001. The standard reference for BN pairs.
Serre, Jean-Pierre (2003). Trees. Springer. ISBN 3-540-44237-5. Zbl 1013.20001.