(B, N) pair

In mathematics, a ("B", "N") pair is a structure on groups of Lie typethat allows one to give uniform proofs of many results, instead of giving a large number ofcase-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.

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 of elements "w_i" of order 2, for "i" in some non-empty set "I".
*If "w_i" is one of the generators of "W" 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 number of generators "w_i" 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 BN 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 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.

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".

The subgroups of "G" containing conjugates of "B" are called parabolic subgroups; conjugates of "B" are called Borel subgroups (or minimal parabolic subgroups). There are exactly 2ⁿ of them containing "B", and they correspond to subsets of "I".

Applications

BN-pairs can be used to prove that most groups of Lie type are simple. 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 or simple). But showing that a group is perfect is usually far easier than showing it is simple.

References

The standard reference for BN pairs is:
* Bourbaki, Nicolas, "Lie Groups and Lie Algebras: Chapters 4-6 (Elements of Mathematics)", ISBN 3-540-42650-7