${\mathrm{GL}}_n$

Let $R$ be a commutative ring. The general linear group ${\mathrm{GL}}_n\left(R\right)$ is

${\mathrm{GL}}_n\left(R\right)=\left\{n\times n\mathrm{ invertible\; matrices\; with\; entries\; in }R\right\}$

Let $𝔽$ be a field.

For $1\le i,j\le n$, $i\ne j$ define

Let $e_1^{\vee },\; ...\; , e_n^{\vee }$ be a $\mathbb{Z}$-basis of $P^{\vee }$ so that

and define

The symmetric group $W=S_n$ acts on $P^{\vee }$ by permuting $e_1^{\vee },\; ...\; , e_n^{\vee }$ and, identifiying $W$ with permutation matrices,

for $w\in W, {\lambda }^{\vee }\in P^{\vee },\; g\in {𝔽}^{\times }$

The group $G{L}_n\left(𝔽\right)$ is presented by generators

$x_{\mathrm{ij}}\left(f\right) and h_{{\lambda }^{\vee }}\left(g\right),1\le i,j\le n, i\ne j, f\in 𝔽, {\lambda }^{\vee }\in P^{\vee }, g\in 𝔽$,

and relations

Introduce a pictorial notation:

	$=$
$s_i$	$=$
$h_{e_i^{\vee }}\left(g\right)$	$=$
so that
$w$	$=$

Subgroups of ${\mathrm{GL}}_n$

$B=\left\{\left(\begin{array}{ccc}*& & *\\ & \ddots & \\ 0& & *\end{array}\right)\right\},T=\left\{\left(\begin{array}{ccc}*& & 0\\ & \ddots & \\ 0& & *\end{array}\right)\right\}$

$U^{*}=\left\{\left(\begin{array}{ccc}1& & *\\ & \ddots & \\ 0& & 1\end{array}\right)\right\},W=\left\{\begin{array}{c}permutation\\ matrices\end{array}\right\}$

$W=\left\{\begin{array}{c}n\times n matrices\; with\hfill \\ (a)\; exactly\; one\; nonzero\; entry\; in\; each\; row\; and\; each\; coloumn\hfill \\ (b)\; nonzero\; entries\; are\; in{𝔽}^{\times }\hfill \end{array}\right\}$

Then

$B=TU,N=W⋉T$

and $U^{+}=\left\{x_{{\beta }_1}\left(c_1\right)···x_{{\beta }_N}\left(c_N\right)\mid c_1,...c_N\in ℂ\right\}$
where$({\beta }_1,...,{\beta }_N)\text{ is a fixed ordering of }\{e_i–e_j\mid 1\le i<j\le n\}$