Is there a root system for a Dihedral group?

Question

Let $G$ be a Coxeter group of type $I_m$ ($m \geq 3$). Then $G$ is a Dihedral group. When $m=3,4,6$, $G$ is of type $A_2, B_2, G_2$ respectively. There are root systems for $A_2, B_2, G_2$. Is there a root system for $I_m$ (for example $m=5$)? If there is one, then we will have two simple roots $\alpha_1,\alpha_2$. What is the ratio $||\alpha_1||/||\alpha_2||$ of the lengths of $\alpha_1,\alpha_2$? Thank you very much.

Answer 1

No, a root system of type $I_m$ only exists for the cases $m = 3,4,6$ you mentioned.

In fact, the irreducible root systems can be classified via Dynkin diagrams and the only possibilities there are $A_n$, $B_n$, $C_n$, $D_n$, $E_6$, $E_7$, $E_8$, $F_4$ and $G_2$.

Answer 2

For the record, there are root systems for all dihedral groups, there simply aren't any crystallographic root systems for dihedral groups with $m\neq 2,3,4,6$. So it depends what definition of root system you're using. For instance neither Kane (Reflection groups and Invariant theory) nor Humphrey (Reflection groups and Coxeter groups) impose the crystallographic condition ($\frac{2(\alpha,\beta)}{(\beta,\beta)}\in \mathbb{Z}$) on root systems.