Past exam of the mathematics course of the University of Cambridge / 2018 / iii / Paper 102 / 3 / a

Solution

H (t) = diag (t_{1}, t_{2}, t_{3}, - t_{1}, - t_{2}, - t_{3}), ε_{i} (H (t)) = t_{i} .

Φ = {\pm ε_{i} \pm ε_{j} : 1 \leq i < j \leq 3} .

Thus the maps requested are

H (t) \mapsto \pm t_{i} \pm t_{j}

, with the two signs independent. This is the D3 root system, with twelve roots.

For a direct check, the defining matrix identity gives the block form

X = (A C B - A^{T})

with

B^{T} = - B

and

C^{T} = - C

. If

E_{ij}

ε_{i} - ε_{j}

is spanned by

E_{ij} - E_{j + 3, i + 3}

; those for

ε_{i} + ε_{j}

and

- ε_{i} - ε_{j}

are spanned respectively by

E_{i, j + 3} - E_{j, i + 3}

and

E_{i + 3, j} - E_{j + 3, i}

. Commuting these with

H (t)

gives exactly the displayed maps.

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