Exterior square of an SU2 irreducible representation

The odd-parity summands give

⋀^{2} V_{n} ≅ ⨁_{k = 0}^{⌊(n - 1) /2 ⌋} V_{2 n - 4 k - 2} .

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- Second and third exterior powers of V4 of SU2 Exterior square of an SU2 irreducible representation

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

The alternating summands in the tensor square give

⋀^{2} V_{4} ≅ V_{6} \oplus V_{2}

. Since

V_{4}

is self-dual and has trivial determinant,

⋀^{3} V_{4} ≅ (⋀^{2} V_{4})^{*} \otimes det V_{4} ≅ V_{6} \oplus V_{2} .

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Past exam of the mathematics course of the University of Cambridge / 2022 / ii / Paper 3 / 19H / Solution

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