(B, N) pair
Encyclopedia
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 invented by the mathematician Jacques Tits
, and are also sometimes known as Tits systems.
The idea of this definition is that B is an analogue of the upper triangular matrices of
the general linear group GLn(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 wi is called the rank.
G = BWB.
The subgroups of G containing conjugates of B are called parabolic subgroups; conjugates of B are called Borel subgroup
s (or minimal parabolic subgroups). There are exactly 2n of them containing B, and they correspond to subsets of I.
, 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.
Mathematics
Mathematics is the study of quantity, space, structure, and change. Mathematicians seek out patterns and formulate new conjectures. Mathematicians resolve the truth or falsity of conjectures by mathematical proofs, which are arguments sufficient to convince other mathematicians of their validity...
, 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
General linear group
In mathematics, the general linear group of degree n is the set of n×n invertible matrices, together with the operation of ordinary matrix multiplication. This forms a group, because the product of two invertible matrices is again invertible, and the inverse of an invertible matrix is invertible...
over a field. They were invented by the mathematician Jacques Tits
Jacques Tits
Jacques Tits is a Belgian and French mathematician who works on group theory and geometry and who introduced Tits buildings, the Tits alternative, and the Tits group.- Career :Tits received his doctorate in mathematics at the age of 20...
, 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 subgroupNormal subgroupIn abstract algebra, a normal subgroup is a subgroup which is invariant under conjugation by members of the group. Normal subgroups can be used to construct quotient groups from a given group....
of N. - The group W = N/H is generated by a set of elements wi of order 2, for i in some non-empty set I.
- If wi is one of the generators of W and w is any element of W, then wiBw is contained in the union of BwiwB and BwB.
- No generator wi normalizes B.
The idea of this definition is that B is an analogue of the upper triangular matrices of
the general linear group GLn(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
Borel subgroup
In the theory of algebraic groups, a Borel subgroup of an algebraic group G is a maximal Zariski closed and connected solvable algebraic subgroup.For example, in the group GLn ,...
, H is sometimes called the Cartan subgroup, and W is called the Weyl group
Weyl group
In mathematics, in particular the theory of Lie algebras, the Weyl group of a root system Φ is a subgroup of the isometry group of the root system. Specifically, it is the subgroup which is generated by reflections through the hyperplanes orthogonal to the roots, and as such is a finite reflection...
.
The number of generators wi 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 GLn(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 monomial matrices, i.e. matrices with exactly one non-zero element in each row and column. There are n − 1 generators wi, represented by the matrices obtained by swapping two adjacent rows of a diagonal matrix
- More generally, any group of Lie typeGroup of Lie typeIn mathematics, a group of Lie type G is a group of rational points of a reductive linear algebraic group G with values in the field k. Finite groups of Lie type form the bulk of nonabelian finite simple groups...
has the structure of a BN-pair.
- A reductive algebraic group over a local fieldLocal fieldIn mathematics, a local field is a special type of field that is a locally compact topological field with respect to a non-discrete topology.Given such a field, an absolute value can be defined on it. There are two basic types of local field: those in which the absolute value is archimedean and...
has a BN-pair where B is an Iwahori subgroupIwahori subgroupIn algebra, an Iwahori subgroup is a subgroup of a reductive algebraic group over a local field that is analogous to a Borel subgroup of an algebraic group. A parahoric subgroup is a subgroup that is a finite union of double cosets of an Iwahori subgroup, so is analogous to a parabolic subgroup of...
.
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 decompositionBruhat decomposition
In mathematics, the Bruhat decomposition G = BWB into cells can be regarded as a general expression of the principle of Gauss–Jordan elimination, which generically writes a matrix as a product of an upper triangular and lower triangular matrices—but with exceptional cases...
G = BWB.
The subgroups of G containing conjugates of B are called parabolic subgroups; conjugates of B are called Borel subgroup
Borel subgroup
In the theory of algebraic groups, a Borel subgroup of an algebraic group G is a maximal Zariski closed and connected solvable algebraic subgroup.For example, in the group GLn ,...
s (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 groupSolvable group
In mathematics, more specifically in the field of group theory, a solvable group is a group that can be constructed from abelian groups using extensions...
, 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
Perfect group
In mathematics, more specifically in the area of modern algebra known as group theory, a group is said to be perfect if it equals its own commutator subgroup, or equivalently, if the group has no nontrivial abelian quotients...
. 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.