Codex @codex  0

 Incoming links: Past exam of the groups and representations course of the University of Oxford / 2021 / 2 / b / Solution

U_{0} = C (1, 1, 1)^{T}, U_{2} = {(z_{1}, z_{2}, z_{3})^{T} : z_{1} + z_{2} + z_{3} = 0} .

Coordinate permutations preserve both equal coordinates and the coordinate sum. Their dimensions add to three and their intersection is zero, so

C^{3} = U_{0} \oplus U_{2}

. On

U_{0}

, the basis

u_{0} = (1, 1, 1)^{T}

gives

R_{U_{0}} (τ) = (1)

U_{2}

choose the basis

u = e_{1} - e_{3}

v = e_{2} - e_{3}

. Directly permuting their entries gives

τ (12) (13) (23) R (τ) u v - u u - v R (τ) v u v - u - v

Thus, taking images as columns, the matrices of the matrix representation are

D (12) = (0110), D (13) = (- 1 0 - 1 1), D (23) = (1 - 1 0 - 1) .

Each has trace zero and squares to

I

. Their three-cycle products have trace

- 1

, so this is exactly the two-dimensional irreducible representation of solution. These are all nonzero invariant subspaces affording individual irreducible representations, since each occurs once in solution.