Past exam of the mathematics course of the year 2 of the natural sciences course of the University of Cambridge / 2021 / Paper 2 / 10 / e / Solution

Taking traces of the given matrices gives the character on the five classes,For a complex finite-group representation, the multiplicity of an irrep $\rho$ is its character inner product with ${\chi }_D$:Every linear character has value one at both $1$ and $-1$, so its multiplicity is $(4-4)/8=0$. For the two-dimensional character,ThereforeThe dimensions check: $2\times 2=4$.

The choice of scalar field matters here, even though the displayed matrices are real. Over $\mathbb{R}$, this particular four-dimensional representation is irreducible: for any nonzero real vector $v$, the vectors $v,D(i)v,D(j)v,D(k)v$ are mutually orthogonal with the same nonzero norm. This follows from the skew-symmetry and quaternion multiplication identities of the three matrices. A nonzero invariant real subspace contains all four vectors and is therefore the whole space. Its complexification is the two-copy decomposition above.