# BN-pair

Let $G$ be a group. Then $G$ has a $BN$-pair or a Tits system if the following conditions hold:

1. 1.

$B$ and $N$ are subgroups of $G$ such that $G=$.

2. 2.

$B\cap N=T\triangleleft N$ and $N/T=W$ is a group generated by a set $S$.

3. 3.

$sBw\subseteq BwB\cup BswB$ for all $s\in S$ and $w\in W$.

4. 4.

$sBs^{-1}\not\subseteq B$ for all $s\in S$.

Where $BwB$ is a double coset with respect to $B$. It can be proven that $S$ is in fact made up of elements of order 2, and that $W$ is a Coxeter group.

Example: Let $G=GL_{n}(\mathbb{K})$ where $\mathbb{K}$ is some field. Then, if we let $B$ be the subgroup of upper triangular matrices and $N$ be the subgroup of monomial matrices (i.e. matrices having one nonzero entry in each row and each column, or more precisely the stabilizer of the lines $\{[e_{1}],...,[e_{n}]\}$). Then, it can be shown that $B$ and $N$ generate $G$ and that $T$ is the subgroup of diagonal matrices. In turn, it follows that $W$ in this case is isomorphic to the symmetric group on $n$ letters, $S_{n}$.

For more, consult chapter 5 in the book Buildings, by Kenneth Brown

Title BN-pair BNpair 2013-03-22 15:30:11 2013-03-22 15:30:11 tedgar (10630) tedgar (10630) 11 tedgar (10630) Definition msc 20F55 Tits Systems