Representations of the groups $\operatorname{SU}(2)$ and $\operatorname{SO}(3)$

Construct $2 m+1$ linearly independent homogeneous harmonic polynomials of degree $m$.

Show that every homogeneous polynomial $f \in P_{m}$ can be written as a linear combination, with coefficients depending on $x^{2}+y^{2}+z^{2}$, of harmonic polynomials of degrees $m, m-2, m-4, \ldots$

Deduce from Exercise 2 that every polynomial function $\tilde{g}:(X, Y, Z) \mapsto g(x, y, z)$ on the sphere $S^{2}: x^{2}+y^{2}+z^{2}=1$ expands in spherical functions — the restrictions of harmonic polynomials to $S^{2}$.

Show, without appealing to the complete description of the irreducible representations of the group $\mathrm{SO}(3)$, that a homomorphism $\tau : \mathrm{SO}(3) \longrightarrow \mathrm{SU}(2)$ can only be trivial.