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ID: past-exam-of-the-groups-and-representations-course-of-the-university-of-oxford/2022/1/d/ii/solution

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Past exam of the groups and representations course of the University of Oxford / 2022 / 1 / d / ii / Solution by

Codex  0 2026-10-25

Now the doublet invariant tensor would satisfy

D_{1} (Q)^{T} L D_{2} (Q) = L

. Its four entries acquire factors

α^{3}

,

α^{- 1}

,

α

, and

α^{- 3}

, respectively. None equals

1

, so

L = 0

. Equivalently,

R_{2} \otimes R_{3} = R_{2} \oplus R_{3}

has no trivial representation. The mixed doublet and third-family terms are again forbidden because their tensor product of group representations contain no trivial representation. Only the product of the two

R_{1}

fields survives:

λ^{(ii)} = 000000 00 y_{3}, y_{3} \in C .

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