Past exam of the mathematics course of the University of Cambridge / 2009 / ii / Paper 3 / 19F / i / 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 $F$,Using the character already found,Thus exactly one irreducible constituent occurs with multiplicity one: $V_n$ is irreducible for every $n\ge 0$.