= Solution
The $n+1$ monomials $z_1^{n-j}z_2^j$ form a <basis>, so $\boxed{\dim V_n=n+1}$. For irreducibility, use the following compact-group facts: every finite-dimensional continuous representation is unitarizable by averaging an <inner product>; such representations decompose into irreducibles; their irreducible characters are orthonormal under normalized Haar measure. In particular the squared character norm is the sum of squares of the irreducible multiplicities.
The <Weyl integration formula for SU2> gives, for a class function $F$,
$$
\int_{SU(2)}F(g)\,dg=\frac2\pi\int_0^\pi F(\theta)\sin^2\theta\,d\theta.
$$
Using the character already found,
$$
\langle\chi_n,\chi_n\rangle=\frac2\pi\int_0^\pi\sin^2((n+1)\theta)\,d\theta=1.
$$
Thus exactly one irreducible constituent occurs with multiplicity one: \b[$V_n$ is irreducible for every $n\ge0$].
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