Past exam of the mathematics course of the University of Cambridge / 2005 / ii / Paper 4 / 19G / ii

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.