(B, N) pair

Last updated August 14, 2024

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, 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 (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

B the (standard) Borel subgroup ,
T the (standard) Cartan subgroup , and
W 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 (or minimal parabolic) if it is a conjugate of B.

Examples

Abstract examples of (B, N) 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 (B, N) pair of rank 1, then the action of G on the cosets of B is doubly transitive. So (B, N) 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 (B, N) pairs can be found in reductive groups.

Suppose that G is the general linear group GL_n K 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(K) has a (B, N) pair in which
- B = P(K), where P is a minimal parabolic subgroup of G, and
- N = N(K), where N is the normalizer of a split maximal torus contained in P.^[2]
In particular, any finite group of Lie type has the structure of a (B, N) 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 (B, N) 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.^[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

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

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References

Abramenko, Peter; Brown, Kenneth S. (2008). Buildings. Theory and Applications. Springer. ISBN 978-0-387-78834-0. MR 2439729. Zbl 1214.20033. Section 6.2.6 discusses BN pairs.
Borel, Armand (1991) [1969], Linear Algebraic Groups, Graduate Texts in Mathematics, vol. 126 (2nd ed.), New York: Springer Nature, doi:10.1007/978-1-4612-0941-6, ISBN 0-387-97370-2, MR 1102012
Bourbaki, Nicolas (1981). Lie Groups and Lie Algebras: Chapters 4–6. Elements of Mathematics (in French). Hermann. ISBN 2-225-76076-4. MR 0240238. Zbl 0483.22001. Chapitre IV, § 2 is the standard reference for BN pairs.
Bourbaki, Nicolas (2002). Lie Groups and Lie Algebras: Chapters 4–6. Elements of Mathematics. Springer. ISBN 3-540-42650-7. MR 1890629. Zbl 0983.17001.
Serre, Jean-Pierre (2003). Trees. Springer. ISBN 3-540-44237-5. Zbl 1013.20001.

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[FOOTNOTEAbramenkoBrown2008319Theorem_6.5.6(1)-1] Abramenko & Brown 2008, p. 319, Theorem 6.5.6(1).

[FOOTNOTEBorel1991236Theorem_21.15-2] Borel 1991, p. 236, Theorem 21.15.

[FOOTNOTEBourbaki198125Théorème_1-3] Bourbaki 1981, p. 25, Théorème 1.

[FOOTNOTEBourbaki198129Théorème_4(iv)-4] Bourbaki 1981, p. 29, Théorème 4(iv).

[FOOTNOTEBourbaki198127Théorème_3-5] Bourbaki 1981, p. 27, Théorème 3.

[FOOTNOTEBourbaki198129Théorème_4-6] Bourbaki 1981, p. 29, Théorème 4.

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