Past exam of the mathematics course of the University of Cambridge / 2005 / ii / Paper 4 / 19G (Representation Theory)

Normalize Haar measure on $\mathrm{SU}(2)$ to total mass one. For a continuous class function $F$, the Weyl integration formula for SU2 isEquivalently it is ${\pi }^{-1}{\int }_0^{2\pi }F(t_{\theta }){\sin }^2\theta d\theta$. For arbitrary continuous $F$, replace $F(t_{\theta })$ on the right by its conjugacy average ${\int }_{\mathrm{SU}(2)}F(h{t}_{\theta }{h}^{-1})dh$.

Here is a direct proof. Identify $\mathrm{SU}(2)$ with unit quaternions, the sphere $S^3$. Multiplication by a unit quaternion is a transformation given by an orthogonal matrix of ${\mathbb{R}}^4$, so normalized round volume is bi-invariant and is Haar measure. Write a quaternion as $\cos \theta +\mathbf{n}\sin \theta$, $0\le \theta \le \pi$, $\mathbf{n}\in S^2$. The round volume element is ${\sin }^2\theta d\theta dA(\mathbf{n})$, and the total volume is $2{\pi }^2$. Conjugation rotates the unit imaginary vector $\mathbf{n}$, so a class function depends only on $\theta$. Integrating over the sphere of area $4\pi$ gives the factor $4\pi /(2{\pi }^2)=2/\pi$. The endpoint degeneracies have measure zero. Conjugacy averaging and Haar invariance prove the more general version.

On the standard two-dimensional representation, $t_{\theta }$ has eigenvalues $e^{i\theta },e^{-i\theta }$. Its $m$th symmetric power has monomial basis $X^{m-j}{Y}^j$, $j=0,\dots ,m$, of weights $m-2j$. The character is thereforeThe quotient is interpreted by continuity at $0,\pi$, where its values are $m+1$ and $(-1{)}^m(m+1)$.

Apply the Weyl integration formula for SU2 to the character product:The last equality follows directly by product-to-sum integration. A finite-dimensional representation of a compact group decomposes into irreducibles, and the squared character norm is the sum of squared multiplicities. Since this norm is one, there is exactly one constituent with multiplicity one. Thus every symmetric power is irreducible, and the different powers are pairwise inequivalent.