SU(2) and SL(2) representations on spaces of homogeneous polynomials

Let $R_{n}$ be the space of homogeneous polynomials of degree $n$ in $x, y$ with complex coefficients. For

and $f \in R_{n}$, set

Prove that the restriction of the representation $\Phi_{n}$ to the subgroup $\mathrm{SU}_{2}(\mathbb {C})$ is irreducible.

Let $R = \mathbb {C}^{x,y}$ be the polynomial ring in two variables $x,y$ over the field $\mathbb {C}$. Let $q$ be a positive integer and $R_{q}$ be the sub vector space of polynomials which are homogeneous of degree $q$. Then $R_{q}$ has the basis

and dimension $q+1$. An element of $SL(2, \mathbb {C})$

operates on $R$ by the transformation $x \to ax+cy$, $y \to bx+dy$ and $R_{q}$ is the representation space for $SL(2, \mathbb {C})$ for each $q$ under this operation. We could also write in matrix form

Let $p_{j} := x^{j}y^{q-j}$, $j = 0, 1, \ldots , q$. Show that the images of $p_{j}$ under the transformation by

are given by

Consider the vector space $V_{2}^{n}$ of homogeneous polynomials of two complex variables. An element of this vector space can be written as

The compact Lie group $SU(2)$ acts on this vector space through the action of $U \in SU(2)$ as a linear transform on the vector $\overrightarrow {z} = (z_{1}, z_{2})^{T}$ by

Show that this a group homomorphism.