(B, N) pair

Concept in group theory 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. Abramenko Brown 2008 319 Theorem 6.5.6(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 GLnK 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. Borel 1991 236 Theorem 21.15 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. Bourbaki 1981 25 Théorème 1 Parabolic subgroups Every parabolic subgroup equals its normalizer in G. Bourbaki 1981 29 Théorème 4(iv) 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. Bourbaki 1981 27 Théorème 3 More generally, this bijection extends to conjugacy classes of parabolic subgroups. Bourbaki 1981 29 Théorème 4 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 References Abramenko Peter Buildings. Theory and Applications Brown Kenneth S. Springer 2008 978-0-387-78834-0 2439729 1214.20033 Section 6.2.6 discusses BN pairs. Borel Armand Armand Borel Linear Algebraic Groups 126 2nd New York Springer Nature 0-387-97370-2 1991 1969 1102012 10.1007/978-1-4612-0941-6 Graduate Texts in Mathematics Nicolas Bourbaki Bourbaki Nicolas Lie Groups and Lie Algebras: Chapters 4–6 Elements of Mathematics Hermann 1981 2-225-76076-4 0240238 0483.22001 French Chapitre IV, § 2 is the standard reference for BN pairs. Nicolas Bourbaki Bourbaki Nicolas Lie Groups and Lie Algebras: Chapters 4–6 Elements of Mathematics Springer 2002 3-540-42650-7 1890629 0983.17001 Trees Jean-Pierre Serre Jean-Pierre Serre Springer 2003 3-540-44237-5 1013.20001 B B B