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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.

For a subgroup $H\le G$ and a left group algebra $k[H]$-module $V$, its induced representation is ${\mathrm{Ind}}_H^{G}V=k[G]{\otimes }_{k[H]}V$, with $G$ acting by left multiplication on the first tensor factor. If $H$ has finite index of a subgroup and $V$ is a finite-dimensional vector space, the dimension of the induced representation is $[G:H]\dim V$.

Inducing the trivial representation gives the permutation representation on cosets $G/H$. Frobenius reciprocity characterizes the construction: intertwining operators from the induced representation to $W$ correspond to $H$-intertwining operators from $V$ to the restriction of a representation $W$. Thus induction of representations extends a smaller group action while imposing only the identifications forced by that subgroup.