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 Incoming links: Past exam of the groups and representations course of the University of Oxford / 2021 / 1 / a / Solution

Past exam of the groups and representations course of the University of Oxford / 2021 / 1 / d / Solution 2026-10-25

The squares of

I, - I, \pm A, \pm B

are

I

, whereas

(\pm C)^{2} = - I

. Therefore

χ_{ρ} (g^{2}) = (2, 2, 2, 2, - 2)

. The symmetric square and exterior square formulas from solution give

χ_{S^{2} ρ} = (3, 3, 1, 1, - 1), χ_{\land^{2} ρ} = (1, 1, - 1, - 1, 1) .

Comparison with the character table gives the Clebsch-Gordan decomposition of group representations

S^{2} ρ = L_{++} \oplus L_{+-} \oplus L_{-+}, \land^{2} ρ = L_{--} .

For explicit bases, put

e_{1} = (1, 0)^{T}, e_{2} = (0, 1)^{T}

. The symmetric tensors

e_{1} \otimes e_{1} + e_{2} \otimes e_{2}

,

e_{1} \otimes e_{2} + e_{2} \otimes e_{1}

, and

e_{1} \otimes e_{1} - e_{2} \otimes e_{2}

transform as

L_{++}, L_{+-}, L_{-+}

respectively. The antisymmetric tensor

e_{1} \otimes e_{2} - e_{2} \otimes e_{1}

L_{--}

, also the determinant representation of

ρ

. Adding these direct sums recovers

ρ \otimes ρ = ⨁_{s, t} L_{s t}

.

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