I am looking at this exercise for 2 days and honestly cannot make any progress, so I really appreciate any help.
Let $V_n$ ⊆ $C[x_1, x_2]$ be the space of all homogeneous polynomials of degree n. Let $Φ_n$: SU2 → $Aut_C(Vn)$ be a map given by
$Φ_n(A)$: $F(x_1, x_2)$ → $F(a_{1,1}x_1+a_{2,1}x_2, a_{1,2}x_1+a_{2,2}x_2)$,
$A$ = \begin{bmatrix}a_{1,1}&a_{1,2}\\a_{2,1}&a_{2,2}\end{bmatrix} ∈ SU2 . Show that $Φ_n$ is irreducible for every n ∈ N.