Coroot

Remarks

With this notation $\langle \beta, \alpha \rangle = (\beta, \alpha^\vee)$, and the reflection in the hyperplane orthogonal to $\alpha$ is $s_\alpha(x) = x - (x, \alpha^\vee) \alpha$. Example: in $B_2 = \set{\pm e_1, \pm e_2, \pm e_1 \pm e_2}$ in $\R^2$, the coroot of $e_1$ is $2 e_1$, and the coroot of $e_1 + e_2$ is $e_1 + e_2$, since $(e_1 + e_2, e_1 + e_2) = 2$. So $\Phi^\vee = \set{\pm 2e_1, \pm 2e_2, \pm e_1 \pm e_2}$, a root system of type $C_2$. Non-example: in $B_2$ the vector $e_1$ is not the coroot of $e_1$, since $(e_1, e_1) = 1$ and so $e_1^\vee = 2e_1$; a root equals its coroot only when $(\alpha, \alpha) = 2$. Many authors see $\alpha^\vee$ instead as the linear form $x \mapsto 2(x, \alpha)/(\alpha, \alpha)$ on $E$, an element of the dual space $E^*$. In general $\Phi^\vee$ is a root system, the dual or inverse root system, and $(\alpha^\vee)^\vee = \alpha$.