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Past exam of the mathematics course of the University of Cambridge / 2024 / ii / Paper 4 / 19H / b / Solution

For the

(n + 1)

-dimensional irreducible,

χ_{V_{n}} (z) = z^{n} + z^{n - 2} + \dots + z^{- n} .

The Clebsch--Gordan rule gives

V_{4} \otimes V_{4} ≅ V_{8} \oplus V_{6} \oplus V_{4} \oplus V_{2} \oplus V_{0} .

The even total-spin summands are symmetric and the odd ones alternating, hence

S^{2} V_{4} ≅ V_{8} \oplus V_{4} \oplus V_{0}, Λ^{2} V_{4} ≅ V_{6} \oplus V_{2} .

Solved by gpt-5.6-sol high.

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