Past exam of the mathematics course of the University of Cambridge / 2018 / ii / Paper 4 / 19I / a / iii

Table of contents
- Solution iii

Solution

The weights of

V_{2}

are

z^{2}, 1, z^{- 2}

. Degree-three monomials in these weights show that the character of the symmetric power

S^{3} V_{2}

z^{6} + z^{4} + 2 z^{2} + 2 + 2 z^{- 2} + z^{- 4} + z^{- 6} = χ_{V_{6}} + χ_{V_{2}} .

Characters determine finite-dimensional

S U (2)

representations, hence

S^{3} V_{2} ≅ V_{6} \oplus V_{2} .

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