Tits system

A collection , where is a group, and are subgroups and is a subset of , satisfying the following conditions: 1) the set generates the group ; 2) is a normal subgroup of ; 3) the set generates the group and consists of elements of order 2; 4) for all , ; and 5) for . The group , called the Weyl group of the Tits system , is a Coxeter group with respect to the system of generators . The correspondence is a bijection from onto the set of double cosets of in .

Examples. a) , where is any field, is the subgroup of upper triangular matrices, is the subgroup of monomial matrices (so that is the subgroup of diagonal matrices and , cf. Monomial matrix), and is the set of transpositions , where .

b) More generally, let be a connected reductive algebraic group over , let be a maximal torus in diagonalizable over , let be its normalizer, let be its centralizer, let be the root system of relative to , let be its Weyl group, and let be the set of reflections corresponding to the simple roots. Moreover, let be the unipotent subgroup of generated by the root subgroups corresponding to the positive roots, and let . Then the quadruple is a Tits system.

c) Let , where is the field of -adic numbers, let be the subgroup consisting of matrices (where is the ring of -adic integers), such that for , and let be the subgroup of monomial matrices. Then there exists a subset such that the quadruple is a Tits system. Here the group is an infinite Coxeter group of type . Analogously one can define Tits systems with Weyl groups of affine type corresponding to other connected reductive groups over local fields.

Under certain conditions one can assert that a group admitting a Tits system is simple. For example, the following conditions are sufficient for this: ) is a solvable group, and is not contained in any proper normal subgroup of ; ) is equal to its own commutator subgroup; ) the Coxeter group is indecomposable; or ) the group does not contain any non-trivial normal subgroup of . In this way one can prove the simplicity of the Chevalley groups, in particular of the finite ones (cf. Chevalley group).

References

[1]	J. Tits, "Buildings of spherical type and finite BN-pairs" , Lect. notes in math. , 386 , Springer (1974)
[2]	N. Bourbaki, "Elements of mathematics. Lie groups and Lie algebras" , Addison-Wesley (1975) pp. Chapt. 4 (Translated from French)

Comments

A Tits system is also called a group with a -pair.

Let be a -transitive permutation group on the set ; then and , where is a permutation in interchanging and , and . This gives a Tits system of type .

References

[a1]	M.A. Ronan, "Lectures on buildings" , Acad. Press (1989)