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Representation theory of SU2

Clebsch-Gordan coefficients

Clebsch-Gordan decomposition for SU2

Flip parity in the SU2 tensor square

Exterior square of an SU2 irreducible representation

{title2=$\bigwedge^2V_4\cong\bigwedge^3V_4\cong V_6\oplus V_2$}

The alternating summands in the tensor square give $\bigwedge^2V_4\cong V_6\oplus V_2$. Since $V_4$ is self-dual and has trivial determinant,
$$
\bigwedge^3V_4\cong(\bigwedge^2V_4)^*\otimes\det V_4
\cong V_6\oplus V_2.
$$


Second and third exterior powers of V4 of SU2

Second and third exterior powers of V4 of SU2 ( $⋀^{2} V_{4} ≅ ⋀^{3} V_{4} ≅ V_{6} \oplus V_{2}$ )

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Second and third exterior powers of V4 of SU2 (⋀2V4​≅⋀3V4​≅V6​⊕V2​)

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Second and third exterior powers of V4 of SU2 ( $⋀^{2} V_{4} ≅ ⋀^{3} V_{4} ≅ V_{6} \oplus V_{2}$ )

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