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 Incoming links: Past exam of the groups and representations course of the University of Oxford / 2018 / 1 / c

Past exam of the groups and representations course of the University of Oxford / 2018 / 1 / d / Solution 2026-10-25

Use the two complementary projection operators

P_{+} = \frac{1}{2} (1111), P_{-} = \frac{1}{2} (1 - 1 - 1 1) .

They obey

P_{+} + P_{-} = I_{2}

,

P_{+} P_{-} = 0

and

P_{\pm}^{2} = P_{\pm}

. The given matrix is

A = P_{+} + α P_{-}

, and consequently

A^{g} = P_{+} + α^{g} P_{-}, A^{3} = I_{2} .

It follows that multiplication of these matrices respects addition modulo three:

R (g) R (h) = A^{g + h} = R (g + h mod 3), R (0) = I_{2} .

Thus this is a group representation. The eigenvectors

(1, 1)^{T}

and

(1, - 1)^{T}

have eigenvalues

1

and

α

, respectively. They span invariant lines carrying

χ_{0}

and

χ_{1}

in the convention of c, so

R ≅ ρ_{0} \oplus ρ_{1}, χ_{R} = (2, 1 + α, 1 + α^{2}) .

The other nontrivial irreducible representation,

ρ_{2}

, is absent.

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