(B, N) pair

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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.

Contents

Definition

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

The set S is uniquely determined by B and N and the pair (W,S) is a Coxeter system . [1]

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 call

A subgroup of G is called

Examples

Abstract examples of (B, N) pairs arise from certain group actions.

More concrete examples of (B, N) pairs can be found in reductive groups.

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. [3]

Parabolic subgroups

Every parabolic subgroup equals its normalizer in G. [4]

Every standard parabolic is of the form BW(X)B for some subset X of S, where W(X) 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. [5] More generally, this bijection extends to conjugacy classes of parabolic subgroups. [6]

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 (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.

Citations

  1. Abramenko & Brown 2008, p. 319, Theorem 6.5.6(1).
  2. Borel 1991, p. 236, Theorem 21.15.
  3. Bourbaki 1981, p. 25, Théorème 1.
  4. Bourbaki 1981, p. 29, Théorème 4(iv).
  5. Bourbaki 1981, p. 27, Théorème 3.
  6. Bourbaki 1981, p. 29, Théorème 4.

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