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

1.
$B$ and $N$ are subgroups^{} of $G$ such that $$.

2.
$B\cap N=T\u25c1N$ and $N/T=W$ is a group generated by a set $S$.

3.
$sBw\subseteq BwB\cup BswB$ for all $s\in S$ and $w\in W$.

4.
$sB{s}^{1}\u2288B$ 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 $G=G{L}_{n}(\mathbb{K})$ 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 $\{[{e}_{1}],\mathrm{\dots},[{e}_{n}]\}$). 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  BNpair 

Canonical name  BNpair 
Date of creation  20130322 15:30:11 
Last modified on  20130322 15:30:11 
Owner  tedgar (10630) 
Last modified by  tedgar (10630) 
Numerical id  11 
Author  tedgar (10630) 
Entry type  Definition 
Classification  msc 20F55 
Synonym  Tits Systems 