Reflection Group

Reflection

A reflection is a linear operator $s=s_{\alpha }$ on $V$ which sends some non-zero vector $\alpha$ to its negative while fixing pointwise the hyperplane $H_{\alpha }$ orthogonal to $\alpha$ with a formula. $s_{\alpha }(\lambda )=\lambda -2\frac{(\alpha ,\lambda )}{(\alpha ,\alpha )}\alpha \forall \lambda \in V$

Finite reflection group

A finite reflection group is a finite group generated by reflections which is a finite subgroup of $O(V)$, we denote as $W$.

Proposition

If $t\in \mathrm{O}(V)$ and $\alpha$ is any nonzero vector in $V$, then $t{s}_{\alpha }{t}^{-1}=s_{t\alpha }$. In particular, if $w\in W$, then $s_{w\alpha }$ belongs to $W$ whenever $s_{\alpha }$ does.

Proof. Obviously $t{s}_{\alpha }{t}^{-1}$ sends $t\alpha$ to its negative. So we need only show that $t{s}_{\alpha }{t}^{-1}$ fixes $H_{t\alpha }$ pointwise. Note that $\lambda$ lies in $H_{\alpha }$ if and only if $t\lambda$ lies in $H_{t\alpha }$, since $(\lambda ,\alpha )=(t\lambda ,ta)$. In turn, $(t{s}_{\alpha }{t}^{-1})(t\lambda )=t{s}_{\alpha }\lambda =t\lambda$ whenever $\lambda$ lies in $H_{\alpha }$.

Rank of Reflection Group

We call the cardinality of simple system the rank of $W$.