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Coxeter group $I_2(n)$ is just a dihedral group $D_{2n}$. This group acts on the plane $\mathbb{R}^2$ and after complexification on $\mathbb{C}^2$, thus it acts on $\mathbb{C}[z_1, z_2]$. Ring of invariants $\mathbb{C}[z_1, z_2]^{D_{2n}}$ is again ring of polynomial on two independent invariants $p_1$ and $p_2$ i.e. $\mathbb{C}[z_1, z_2]^{D_{2n}}=\mathbb{C}[p_1, p_2]$. Could someone explain to me (or give a reference) how one can get an equation of the discriminant(=image of all irregular orbits) in the orbit space?

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    In general, Humphreys' *Reflection groups and Coxeter groups* is a good reference for anything pertaining to Coxeter groups2012-08-05

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The discriminant is the image in the orbit space of the union of the "mirrors" of the reflections in the dihedral group. That hypersurface is defined by $Q(z_1,z_2)$, the product of the corresponding linear forms. One identifies the ambient space ${\mathbb C}^2/D_{2n}$ with ${\mathbb C}^2$ via the map $(z_1,z_2) \mapsto (x_1,x_2)=(p_1(z_1,z_2),p_2(z_1,z_2))$. Thus one needs a polynomial that vanishes on the image of $Q=0$ under this map. The square $Q^2$ of $Q$ in an invariant polynomial, hence is expressible uniquely as a polynomial in $p_1$ and $p_2$, i.e., $F(p_1,p_2)= Q(z_z,z_2)^2$. The polynomial $F(x_1,x_2)$ is then the defining polynomial of the discriminant.

For example, for the (reducible) Coxeter group with one reflection across the line $z_1-z_2=0$, and basic invariants $p_1=z_1+z_2$ and $p_2=z_1z_2$, one has $(z_1-z_2)^2=p_1^2-4z_2$, so the discriminant is defined by $x_1^2-4x_2$.