BN-pair

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

1.
$B$ and $N$ are subgroups of $G$ such that $G=<B,N>$ .
2.
$B\\cap N=T\\triangleleft N$ 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.
$sBs^{-1}\ot\\subseteq B$ for all $s\\in S$ .

Where $B w B$ 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

单词	BNpair
释义	BN-pair Let $G$ be a group. Then $G$ has a $B N$ -pair or a Tits system if the following conditions hold: 1. $B$ and $N$ are subgroups of $G$ such that $G=<B,N>$ . 2. $B\\cap N=T\\triangleleft N$ 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. $sBs^{-1}\ot\\subseteq B$ for all $s\\in S$ . Where $B w B$ 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
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