Past exam of the mathematics course of the University of Cambridge / 2018 / ii / Paper 4 / 19I (Representation Theory)

The special unitary group in two dimensions isLet $V_1={\mathbb{C}}^2$ be its defining group representation, and for $n\ge 0$ letthe homogeneous polynomial representation of SU2 on homogeneous polynomials of degree $n$ in two variables. It has degree of a representation $n+1$. The representations $V_0,V_1,V_2,\dots$ are pairwise nonisomorphic, continuous, finite-dimensional irreducible representations, and the classification of finite-dimensional representations of SU2 says that this is the complete list.

Every $A\in SU(2)$ is conjugate to $\mathrm{diag}(z,z^{-1})$ with $|z|=1$. The monomial basis of $V_n$ gives the character of the homogeneous polynomial representation of SU2Writing $z=e^{i\theta }$, this is $\sin ((n+1)\theta )/\sin \theta$, interpreted by continuity when $\sin \theta =0$. The Clebsch-Gordan decomposition for SU2 is

Applying the Clebsch-Gordan decomposition for SU2 givesandOn the summand $V_{6-2j}$, interchange of the two tensor factors acts by $(-1{)}^j$; this is the flip parity in the SU2 tensor square. The even-$j$ summands make the symmetric square, so

Repeatedly using $V_r\otimes V_1\cong V_{r+1}\oplus V_{r-1}$ shows that the multiplicity of $V_{n-2j}$ in the nth tensor power satisfies Pascal's recursion. Consequentlywhere $\left(\genfrac{}{}{0ex}{}{n}{-1}\right)=0$. The multiplicity is the difference between the numbers of weight vectors of weights $n-2j$ and $n-2j+2$.

The weights of $V_2$ are $z^2,1,z^{-2}$. Degree-three monomials in these weights show that the character of the symmetric power $S^3{V}_2$ has weightsCharacters determine finite-dimensional $SU(2)$ representations, hence

Writing $\rho (A)X=A_{1}X{A}_1^{-1}$, we haveand $\rho (I)=I$, so this conjugation representation is indeed a group representation.

The underlying three-dimensional module isUsing $M_3(\mathbb{C})=\mathrm{End}(W)\cong W\otimes W^{\ast }$, the self-duality of finite-dimensional SU2 representations, and $V_1\otimes V_1\cong V_2\oplus V_0$, we obtainTherefore the irreducible decomposition is