= Symmetric powers of the three-dimensional SU2 representation
{title2=$\operatorname{Sym}^n(V_2)=\bigoplus_{j=0}^{\lfloor n/2\rfloor}V_{2n-4j}$}
The three-dimensional irreducible <representation> $V_2$ of <SU(2)> has weights $2,0,-2$. Thus the <character> of its $n$th <symmetric power> is $\sum_{a+b+c=n}t^{2(a-c)}$. Its weight $2\ell$ has multiplicity $\lfloor(n-|\ell|)/2\rfloor+1$ for $|\ell|\leq n$. Subtracting adjacent weight multiplicities gives one copy of each $V_{2n},V_{2n-4},\ldots$. The invariant subspace is one-dimensional for even $n$ and zero for odd $n$.
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