(B, N) pair

A Tits system (often synonymously called BN- pair) is used in the mathematical discipline of group theory to formulate many results from the theory of semisimple Lie groups, algebraic groups and finite groups of Lie - type uniform and prove to can. In addition, the Tits - systems are the algebraic counterpart of building theory. The term was introduced by Jacques Tits.

Definition

A Tits system consists of a 4 -tuple, with one group, and subgroups of are and a lot of cosets of in is so that the following four axioms are satisfied:

The numbering T1 to T4 comes from Tits ' original work.

Examples

• It is often given over a field K as a standard example, the group of invertible matrices. Here, B is the subset of the upper triangular matrices. For the group N we take all matrices that have exactly one entry in each row and in each column equal to zero. The group is then exactly to the group of diagonal matrices and is canonically isomorphic to the symmetric group on n elements. The set S consists of the permutations that exchange two adjacent elements.
• Let G be a reductive algebraic general group and B a Borel subgroup containing a maximal torus H. Let N be the normalizer of H in G and W is a minimal system of generators of W: = N / H. Then (G, B, N, S ) is a tits system.
• Let X be a set with at least three elements and G is a subgroup of the permutation group of X, so that G acts on X doubly transitive. Furthermore, two different elements are given. Then B is the Stabilistator of x in G, and N is defined as the group that holds the amount of the amount, ie the elements X and Y are either both fixed or reversed. Then arises as a pointwise stabilizer of the crowd. The factor group W: = N / H has order 2 and the set S consists of only one element, and this corresponds to the interchange of x and y.

Comments

One can show that the set S is uniquely determined if a Tits system, only the groups G, B, N are given. Moreover, since the group G of B and N is generated, puts all the information about the Tits system in groups B and N. hence the name BN- pair has become the norm.

Bruhat decomposition

An important result which can be proved in the general framework of Tits systems, is the so-called Bruhat decomposition: If a Tits system (G, B, N, S ) is given, then applies

,

With a disjoint union, that is, is chosen so that for the quantities BwB and Bw'B are disjoint.

Applications

If a Tits system (G, B, N, S ) nor the following additional properties are satisfied:

• B is solvable
• The intersection of all the conjugates of B is trivial
• The set S can not be decomposed into two disjoint subsets nichtkommutierende
• G is perfect

Then the group G is a simple group. Often it is very easy to verify the first three properties, and it only remains the perfectness of G to show what is much easier than to show directly that G is a simple group. This result is used for example in the classification of finite simple groups, in order to show that most finite groups of Lie type are simple.

Connection with building theory

It is often helpful to examine groups, by allowing it to act on interesting geometric objects. Each Tits system (G, B, N, S) can be attributed to canonical way assign a geometric object, called building so that G acts on this building. Conversely, each building also assign a Tits system, so that the group-theoretical theory of Tits systems in some way equivalent to the geometric theory of building.