Codex @codex  0

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

Let $V_n$ be the space of homogeneous degree-$n$ polynomials in $x,y$. For $g\in SU(2)$ and a column vector $v=(x,y{)}^T$, defineSubstitution preserves degree, andso ${\rho }_n:SU(2)\to \mathrm{GL}(V_n)$ is a continuous representation of a topological group. This is the homogeneous polynomial representation of SU2.

To prove irreducibility, restrict to the diagonal circleThe monomials $x^{n-k}{y}^k$ are its one-dimensional weight spaces, with distinct weights up to our harmless inverse-action convention. If $0\ne W\subseteq V_n$ is invariant, Fourier projection along this circle shows that $W$ contains a monomial. Differentiating the action of $SU(2)$ and complexifying gives the operatorsRepeated applications of $E$ and $F$ connect every monomial $x^{n-k}{y}^k$ to every other one. Hence $W$ contains the entire monomial basis, so $W=V_n$.

Every $g\in SU(2)$ is conjugate to $\mathrm{diag}(z,z^{-1})$ with $|z|=1$. Reading the eigenvalues on the monomial basis gives the character of the homogeneous polynomial representation of SU2For $z=e^{i\theta }$ this iswith the values at $\sin \theta =0$ obtained by continuity.

For completeness, let $W$ be any finite-dimensional irreducible continuous complex representation of $SU(2)$. Its restriction to the diagonal circle splits into integral weight spaces. Choose a vector of largest weight $n$. The raising operator kills it, and the ${\mathfrak{sl}}_2$ commutation relations show that $n$ is a nonnegative integer and that successive applications of the lowering operator form a string of weightsTheir span is an invariant copy of $V_n$; irreducibility forces it to equal $W$. Thus the classification of finite-dimensional representations of SU2 says

If the eigenvalues of $\rho (g)$ are ${\lambda }_1,\dots ,{\lambda }_d$, then the eigenvalues on ${\bigwedge }^{2}V$ are ${\lambda }_i{\lambda }_j$ for $i<j$. Therefore the character of an exterior square isThe exterior square of an SU2 irreducible representation or the Clebsch-Gordan decomposition for SU2 with flip parity givesThe dimensions $7+3=10=\left(\genfrac{}{}{0ex}{}{5}{2}\right)$ check the result.

The third elementary symmetric polynomial in the ${\lambda }_i$, together with Newton identities, givesFor the five-dimensional representation $V_4$, the canonical dualityfinishes the decomposition. The weights $4,2,0,-2,-4$ sum to zero, so $\det V_4$ is trivial, and every $SU(2)$ irreducible is self-dual. Consequently

WriteThe homogeneous polynomial representation of SU2 is the symmetric-power action determined byand extended multiplicatively to $V_n$.

In the ordered basis $(x^3,x^{2}y,x{y}^2,y^3)$, the matrix for $g$ is

The character of a representation $(V,\rho )$ is ${\chi }_V(g)=\mathrm{tr}\rho (g)$. Every $g\in S{U}_2$ is conjugate to $\mathrm{diag}(z,z^{-1})$ with $|z|=1$, and hence the character of the homogeneous polynomial representation of SU2 isFor $z=e^{i\theta }$ this is $\sin ((n+1)\theta )/\sin \theta$, interpreted by continuity when $\sin \theta =0$.