 Exterior square of an SU2 irreducible representation

For the irreducible representation $V_n={\mathrm{Sym}}^n({\mathbb{C}}^2)$ of SU(2), the exterior square decomposes asFor $n=0$, the sum is empty and the exterior square is zero.

The Clebsch-Gordan decomposition for SU2 first gives $V_n\otimes V_n\cong {⨁}_{j=0}^n{V}_{2n-2j}$. By flip parity in the SU2 tensor square, exchanging the two tensor factors acts on the $j$th summand by $(-1{)}^j$. The exterior square is the minus-one eigenspace of this exchange, so it selects exactly the odd indices $j=2k+1$. Substitution gives the displayed formula.

For example, ${\bigwedge }^2{V}_2\cong V_2$ has dimension three, while ${\bigwedge }^2{V}_3\cong V_4\oplus V_0$ has dimension $5+1=6$, as required for the exterior square of a four-dimensional vector space.