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.
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 BN 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 BN 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 BN pairs can be found in reductive groups.
Suppose that G is the general linear group GLn(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 BN 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]
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.
Bourbaki, Nicolas (1981). Lie Groups and Lie Algebras: Chapters 4–6. Elements of Mathematics (in French). Hermann. ISBN2-225-76076-4. MR 0240238. Zbl 0483.22001. Chapitre IV, § 2 is the standard reference for BN pairs.
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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 1 Definition 1 1 Terminology 2 Examples 3 Properties 3 1 Bruhat decomposition 3 2 Parabolic subgroups 3 3 Tits s simplicity theorem 4 Citations 5 ReferencesDefinition EditA 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 Edit 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 EditAbstract examples of BN 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 BN 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 BN pairs can be found in reductive groups Suppose that G is the general linear group GLn 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 BN 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 BN 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 BN pair where B is an Iwahori subgroup Properties EditBruhat decomposition Edit 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 Edit 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 Edit 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 Edit Abramenko amp Brown 2008 p 319 Theorem 6 5 6 1 Borel 1991 p 236 Theorem 21 15 Bourbaki 1981 p 25 Theoreme 1 Bourbaki 1981 p 29 Theoreme 4 iv Bourbaki 1981 p 27 Theoreme 3 Bourbaki 1981 p 29 Theoreme 4 References EditAbramenko 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 Retrieved from https en wikipedia org w index php title B N pair amp oldid 1109956253, wikipedia, wiki, book, books, library,
