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 Incoming links: Past exam of the groups and representations course of the University of Oxford / 2024 / 1 / c

Past exam of the groups and representations course of the University of Oxford / 2024 / 1 / d / Solution 2026-10-25

All three displayed transposition matrices have trace zero. Multiplying two of them gives

R (τ_{1} τ_{2}) = diag (α, α^{- 1}), R (τ_{2} τ_{1}) = diag (α^{- 1}, α),

since

α^{3} = 1

. Their traces are

α + α^{- 1} = - 1

, whereas

tr R (e) = 2

. Therefore

χ_{R} = (2, 0, - 1), ⟨ χ_{R}, χ_{R} ⟩ = \frac{4 + 3 \cdot 0 + 2 \cdot 1}{6} = 1.

By character orthogonality,

R

is the irreducible representation

2

from c.

There is also a direct invariant subspace argument. The two distinct eigenvalues of

R (τ_{1} τ_{2})

force any invariant line to be one of the two coordinate lines. But

R (τ_{3})

exchanges those lines. Neither is invariant under the whole group, so this two-dimensional representation is irreducible.

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