Tits system

2020 Mathematics Subject Classification: Primary: 20E Secondary: 20G [MSN][ZBL]

A Tits system is a collection $(G,B,N,S)$, where $G$ is a group, $B$ and $N$ are subgroups and $S$ is a subset of $N/(B\cap N)$, satisfying the following conditions:

1) the set $B\cap N$ generates the group $G$;

2) $T=B\cap N$ is a normal subgroup of $N$;

3) the set $S$ generates the group $W = N/T$ and consists of elements of order 2;

4) $sBw\subset BwB\cap BswB$ for all $s\in S$, $w\in W$; and

5) $sBs\not\subset B$ for $s\in S$.

The group $W$, called the Weyl group of the Tits system $(G,B,N,S)$, is a Coxeter group with respect to the system of generators $S$. The correspondence $w\mapsto BwB$ is a bijection from $W$ onto the set of double cosets of $B$ in $G$.

Examples. a) $G = \def\GL{\textrm{GL}}\GL_n(k)$, where $k$ is any field, $B$ is the subgroup of upper triangular matrices, $N$ is the subgroup of monomial matrices (so that $T$ is the subgroup of diagonal matrices and $W = S_n$, cf. Monomial matrix), and $S$ is the set of transpositions $(i\; i+1)$, where $i=1,\dots,n-1$.

b) More generally, let $G$ be a connected reductive algebraic group over $k$, let $T$ be a maximal torus in $G$ diagonalizable over $k$, let $N$ be its normalizer, let $Z$ be its centralizer, let $R$ be the root system of $G$ relative to $T$, let $W = N/Z$ be its Weyl group, and let $S$ be the set of reflections corresponding to the simple roots. Moreover, let $U$ be the unipotent subgroup of $G$ generated by the root subgroups corresponding to the positive roots, and let $P = UZ$. Then the quadruple $(G(k),P(k),N(k),S)$ is a Tits system.

c) Let $G = GL_n(\Q_p)$, where $\Q_p$ is the field of $p$-adic numbers, let $B$ be the subgroup consisting of matrices $(a_{ij})\in \GL_n(\Z_p)$ (where $\Z_p$ is the ring of $p$-adic integers), such that $a_{ij}\in p\Z_p$ for $i>j$, and let $N$ be the subgroup of monomial matrices. Then there exists a subset $S\subset W = N/(B\cap N) $ such that the quadruple $(G,B,N,S)$ is a Tits system. Here the group $W$ is an infinite Coxeter group of type $\tilde A_{n-1}$. 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 $G$ admitting a Tits system is simple. For example, the following conditions are sufficient for this:

$\alpha$) $B$ is a solvable group, and is not contained in any proper normal subgroup of $G$;

$\beta$) $G$ is equal to its own commutator subgroup;

$\gamma$) the Coxeter group $W$ is indecomposable; or

$\delta$) the group $B$ does not contain any non-trivial normal subgroup of $G$.

In this way one can prove the simplicity of the Chevalley groups, in particular of the finite ones (cf. Chevalley group).

Comments

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

Let $G$ be a $2$-transitive permutation group on the set $\Omega=\{1,2,\dots\}$; then $S = \{s\}$ and $W=\{1,s\}$, where $s$ is a permutation in $G$ interchanging $1$ and $2$, $B=G_1$ and $N=G_{\{1,2\}}$. This gives a Tits system of type $A_1$.

References

[Bo]	N. Bourbaki, "Elements of mathematics. Lie groups and Lie algebras", Addison-Wesley (1975) pp. Chapt. 4 (Translated from French) MR0240238 Zbl 1145.17001
[Ro]	M.A. Ronan, "Lectures on buildings", Acad. Press (1989) MR1005533 Zbl 0694.51001
[Ti]	J. Tits, "Buildings of spherical type and finite BN-pairs", Lect. notes in math., 386, Springer (1974) MR0470099 Zbl 0295.20047