Representation theory of SU2

Finite-dimensional complex representations of $S{U}_2$ are completely reducible, and their irreducibles are indexed by nonnegative integers.

The irreducible representation $V_n={\mathrm{Sym}}^n({\mathbb{C}}^2)$ consists of homogeneous degree-$n$ polynomials in two variables, with the action induced from the standard two-dimensional representation. It has dimension $n+1$.

If $g\in S{U}_2$ has eigenvalues $z,z^{-1}$, then

Every finite-dimensional complex $S{U}_2$-representation is a direct sum of the pairwise nonisomorphic irreducibles $V_n={\mathrm{Sym}}^n({\mathbb{C}}^2)$.

The standard invariant alternating form identifies $V_1$ with its dual. Its symmetric powers identify every $V_n$ with $V_n^{\ast }$, and complete reducibility then gives $V\cong V^{\ast }$ for every finite-dimensional complex $S{U}_2$-representation.

On $V_n$, the central element $-I$ acts as $(-1{)}^n$. It therefore acts trivially on $V_n\otimes V_n$, although it need not act trivially on the tensor square of a representation containing irreducibles of both parities.

Clebsch-Gordan coefficients are the change-of-basis coefficients between an uncoupled tensor-product basis and a basis adapted to the irreducible decomposition of the tensor product.

For $m,n\ge 0$,The identity follows by multiplying the weight characters and comparing their nested weight strings.

On the multiplicity-one summand $V_{2n-2j}\subset V_n\otimes V_n$, interchange of tensor factors acts by $(-1{)}^j$. A highest-weight vector exhibiting the sign is

The odd-parity summands give

The alternating summands in the tensor square give ${\bigwedge }^2{V}_4\cong V_6\oplus V_2$. Since $V_4$ is self-dual and has trivial determinant,