= Solution
Taking traces of the given matrices gives the <character> on the five classes,
$$
\chi_D=(4,-4,0,0,0).
$$
For a complex finite-group representation, the multiplicity of an irrep $\rho$ is its <character inner product> with $\chi_D$:
$$
n_\rho=\frac1{|H|}\sum_{C}|C|\,\chi_\rho(C)^*\chi_D(C).
$$
Every linear <character> has value one at both $1$ and $-1$, so its multiplicity is $(4-4)/8=0$. For the two-dimensional <character>,
$$
n_2=\frac18[2\cdot4+(-2)(-4)]=2.
$$
Therefore
$$
\boxed{D\cong\rho_2\oplus\rho_2,\qquad(n_{++},n_{+-},n_{-+},n_{--},n_2)=(0,0,0,0,2).}
$$
The dimensions check: $2\times2=4$.
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.
Back to article page
