(B, N) pair

In mathematics, a 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 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, T, of B and N is a normal subgroup of N.
The group W = N/''T is generated by a set S'' of elements of order 2 such that
*If s is an element of S and w is an element of W then sBw is contained in the union of BswB and BwB.
*No element of S normalizes B.

The set S is uniquely determined by B and N and the pair is a Coxeter system.

Terminology

BN pairs are closely related to reductive groups and the terminology in both subjects overlaps. The size of S is called the rank. We callB the Borel subgroup,T the Cartan subgroup, andW the Weyl group.
A subgroup of G is called

parabolic if it contains a conjugate of B,
standard parabolic if, in fact, it contains B itself, and
a Borel if it is a conjugate of B.

Examples

Abstract examples of pairs arise from certain group actions.

Suppose that G is any doubly transitive permutation group on a set E 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 T 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 pair of rank 1, then the action of G on the cosets of B is doubly transitive. So pairs of rank 1 are more or less the same as doubly transitive actions on sets with more than 2 elements.

More concrete examples of pairs can be found in reductive groups.

Suppose that G is the general linear group GL_nK over a field K. We take B to be the upper triangular matrices, T 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, represented by the matrices obtained by swapping two adjacent rows of a diagonal matrix. The Weyl group is the symmetric group on n letters.
More generally, if G is a reductive group over a field K then the group G = G has a pair in which
* B = P, where P is a minimal parabolic subgroup of G, and
*N = N, where N is the normalizer of a split maximal torus contained in P.
In particular, any finite group of Lie type has the structure of a pair.
*Over the field of two elements, the Cartan subgroup is trivial in this example.
A semisimple simply-connected algebraic group over a local field has a pair where B is an Iwahori subgroup.

Properties

Bruhat decomposition

The Bruhat decomposition states that G = BWB. More precisely, the double cosets B\G/B are represented by a set of lifts of W to N.

Parabolic subgroups

Every parabolic subgroup equals its normalizer in G.
Every standard parabolic is of the form BW''B'' for some subset X of S, where W denotes the Coxeter subgroup generated by X. Moreover, two standard parabolics are conjugate if and only if their sets X are the same. Hence there is a bijection between subsets of S and standard parabolics. More generally, this bijection extends to conjugacy classes of parabolic subgroups.

Tits's simplicity theorem

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. But showing that a group is perfect is usually far easier than showing it is simple.