 Solution

ID: past-exam-of-the-groups-and-representations-course-of-the-university-of-oxford/2021/2/b/solution

σ, τ

, the prescribed linear maps satisfy

R (σ) R (τ) e_{k} = R (σ) e_{τ (k)} = e_{σ (τ (k))} = R (σ τ) e_{k} .

Since this holds on a basis,

R (σ) R (τ) = R (σ τ)

. Also

R (e) = I

and

R (σ)^{- 1} = R (σ^{- 1})

. Hence

R

The trace of a permutation matrix counts its fixed coordinate basis vectors. There are respectively

3, 1, 0

fixed vectors on the three conjugacy classes, so

χ_{R} = (3, 1, 0) = χ_{1} + χ_{D}

. Consequently

R = 1 \oplus D .

In particular the sign representation is absent.

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