Tits System

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.

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, ''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.

Terminology

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

Abstract examples of (''B'', ''N'') 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 (''B'', ''N'') 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 (''B'', ''N'') pairs can be found in reductive groups.
*Suppose that ''G'' is the general linear group GL_''n''''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 (''B'', ''N'') 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.
*In particular, any finite group of Lie type has the structure of a (''B'', ''N'') 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 (''B'', ''N'') pair where ''B'' is an Iwahori subgroup.

Properties

Bruhat decomposition

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''.

Parabolic subgroups

Every parabolic subgroup equals its normalizer in ''G''.

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. More generally, this bijection extends to conjugacy classes of parabolic subgroups.

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.

Citations

References

* Section 6.2.6 discusses BN pairs.
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* Chapitre IV, § 2 is the standard reference for BN pairs.
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* {{cite book , title=Trees , first=Jean-Pierre , last=Serre , authorlink=Jean-Pierre Serre , publisher=Springer , year=2003 , isbn=3-540-44237-5 , zbl=1013.20001

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