Past exam of the mathematics course of the University of Cambridge / 2009 / ii / Paper 3 / 19F (Representation Theory)

The left group action on the homogeneous polynomials is ${\rho }_n(g)P(z)=P(g^{-1}z)$, since ${\rho }_n(g){\rho }_n(h)={\rho }_n(gh)$. Every SU(2) matrix is conjugate to $\mathrm{diag}(e^{i\theta },e^{-i\theta })$ with $0\le \theta \le \pi$. In the monomial basis $z_1^{n-j}{z}_2^j$, its eigenvalues are $e^{-i(n-2j)\theta }$. Consequently the character of the homogeneous polynomial representation of SU2 isThe quotient is interpreted by continuity at the endpoints: ${\chi }_n(0)=n+1$ and ${\chi }_n(\pi )=(-1{)}^n(n+1)$. The introductory action and character requests are independent of parts (i)–(iii), so this root Solution slot has been restored.

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 $n\ge 0$.

The same Weyl integration formula for SU2 gives $⟨{\chi }_n,{\chi }_m⟩={\delta }_{nm}$. More strongly, these characters are complete in the space of square-integrable class functions: multiplication by $\sin \theta$ identifies that weighted space with $L^2(0,\pi )$, and sends ${\chi }_n$ to $\sin ((n+1)\theta )$. The sine functions form a complete orthogonal basis of $L^2(0,\pi )$, as follows from the Fourier series theorem applied to odd extensions.

Suppose a finite-dimensional continuous irreducible group representation were not isomorphic to any $V_n$. Character orthogonality would make its character orthogonal to every ${\chi }_n$. Completeness forces its character to vanish in $L^2$, and continuity then forces it to vanish everywhere. This is impossible at the identity, where the character equals the positive dimension of the representation. Hence $\boxed{\text{every such irreducible representation is some }{V}_n}$. Since the dimensions $n+1$ differ, the index is unique.

The character of the dual representation is ${\chi }_{V_n^{\ast }}(g)={\chi }_n(g^{-1})=\overline{{\chi }_n(g)}$, using unitarizability. The displayed sine quotient is real, so this character equals ${\chi }_n$. The theorem that characters determine finite-dimensional semisimple complex representations of a compact group therefore gives $\boxed{V_n\cong V_n^{\ast }}$. This is complex-linear self-duality; it does not assert that every $V_n$ has an invariant real structure.