Tits system

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

definition

A Tits system consists of a 4-tuple , where a group is and subgroups of are and a set of subclasses is of in such that the following four axioms are satisfied: ${\displaystyle (G,B,N,S)}$ ${\displaystyle G}$ ${\displaystyle B}$ ${\displaystyle N}$ ${\displaystyle G}$ ${\displaystyle S\subseteq N/(B\cap N)}$ ${\displaystyle B\cap N}$ ${\displaystyle N}$

T 1:	The group is created by and . There is also a normal divisor in . ${\displaystyle G}$ ${\displaystyle B}$ ${\displaystyle N}$ ${\displaystyle H:=B\cap N}$ ${\displaystyle N}$
T 2:	The factor group is generated by the crowd and it applies to everyone . ${\displaystyle W:=N/H}$ ${\displaystyle S}$ ${\displaystyle s^{2}=1}$ ${\displaystyle s\in S}$
T 3:	For and applies . ${\displaystyle s\in S}$ ${\displaystyle w\in W}$ ${\displaystyle sBw\subseteq BswB\cup BwB}$
T 4:	For is not a subset of . ${\displaystyle s\in S}$ ${\displaystyle sBs^{-1}}$ ${\displaystyle B}$

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

Examples

• The group of invertible matrices ${\displaystyle G:=\mathrm {GL} (n,K)}$ ${\displaystyle n\times n}$ over a body is often given as a standard example . Here is the subgroup of the upper triangular matrices . For the group we take all matrices that have exactly one entry not equal to zero in each row and in each column. The group then becomes exactly the group of the diagonal matrices and is canonically isomorphic to the symmetric group over elements . The set consists of the permutations that exchange two neighboring elements. ${\displaystyle K}$ ${\displaystyle B}$ ${\displaystyle N}$ ${\displaystyle H:=B\cap N}$ ${\displaystyle W=N/H}$ ${\displaystyle n}$ ${\displaystyle S}$
• More generally, let be a reductive algebraic group and a Borel subgroup that contains a maximal torus . Let be the normalizer of in and a minimal generating system of . Then there is a tits system. ${\displaystyle G}$ ${\displaystyle B}$ ${\displaystyle H}$ ${\displaystyle N}$ ${\displaystyle H}$ ${\displaystyle G}$ ${\displaystyle S}$ ${\displaystyle W:=N/H}$ ${\displaystyle (G,B,N,S)}$
• Let be a set with at least three elements and a subgroup of the permutation group of , so that acts twice transitive on . Furthermore, two different elements are given. Then let be the stabilizer of in and be defined as the group that fixes the quantity as a quantity, i.e. H. the elements and are either both fixed or swapped. The amount then results as a point-by-point stabilizer . The factor group has order 2 and the set consists only of a single element and this corresponds to the exchange of and . ${\displaystyle X}$ ${\displaystyle G}$ ${\displaystyle X}$ ${\displaystyle G}$ ${\displaystyle X}$ ${\displaystyle x,y\in X}$ ${\displaystyle B}$ ${\displaystyle x}$ ${\displaystyle G}$ ${\displaystyle N}$ ${\displaystyle \{x,y\}}$ ${\displaystyle x}$ ${\displaystyle y}$ ${\displaystyle H:=B\cap N}$ ${\displaystyle \{x,y\}}$ ${\displaystyle W:=N/H}$ ${\displaystyle S}$ ${\displaystyle x}$ ${\displaystyle y}$

Remarks

One can show that the set is clearly defined if only the groups of a Tits system are given. In addition, since the group is created by and , all information about the Tits system is contained in groups and . That is why the designation BN pair has become common. ${\displaystyle S}$ ${\displaystyle G,B,N}$ ${\displaystyle G}$ ${\displaystyle B}$ ${\displaystyle N}$ ${\displaystyle B}$ ${\displaystyle N}$

Bruhat decomposition

An important result that can be proven in the general framework of Tits systems is the so-called Bruhat decomposition: If a Tits system is given, then the following applies ${\displaystyle (G,B,N,S)}$

${\displaystyle G=BNB=\bigcup _{n\in N}BnB=\bigcup _{w\in W}BwB}$,

where is a disjoint union, i.e. is chosen such that for the sets and are disjoint. ${\displaystyle \bigcup _{w\in W}BwB}$ ${\displaystyle W}$ ${\displaystyle w\neq w'}$ ${\displaystyle BwB}$ ${\displaystyle Bw'B}$

Applications

If the following additional properties are also fulfilled in a Tits system : ${\displaystyle (G,B,N,S)}$

• ${\displaystyle B}$is resolvable
• The intersection of all conjugates of is trivial ${\displaystyle B}$
• The set cannot be broken down into two disjoint, non-commutating subsets ${\displaystyle S}$
• ${\displaystyle G}$is perfect

Then the group is a simple group. Often it is very easy to check the first three properties and all that remains is to show the perfection of , which is much easier than directly showing that it is a simple group. This result is used, for example, in the classification of the simple finite groups to show that most finite groups of the Lie type are simple. ${\displaystyle G}$ ${\displaystyle G}$ ${\displaystyle G}$

Connection with building theory

It is often helpful to examine groups by letting them act on interesting geometric objects. A geometric object, called a building, can be assigned to each Tits system in a canonical way, so that it acts on this building. Conversely, a Tits system can also be assigned to each building, so that the group-theoretical theory of the Tits systems is in a certain way equivalent to the geometric theory of buildings. ${\displaystyle (G,B,N,S)}$ ${\displaystyle G}$

Web links

• Jacques Tits: Formes quadratiques, groupes orthogonaux et algèbres de Clifford. Inventiones Mathematicae 1968