Past exam of the groups and representations course of the University of Oxford / 2020 / 1 / b / Solution

Write $P=(P_{ij}{)}_{i,j=1}^3$, where $P_{ij}:{\mathbb{C}}^{m_j}\to {\mathbb{C}}^{m_i}$. Block multiplication makes the commuting condition equivalent toThus every block is an intertwiner. For $i\ne j$, the irreducible representations are inequivalent, so the Schur lemma gives $P_{ij}=0$. On each diagonal block, the same Schur lemma gives $P_{ii}={\lambda }_i{I}_{m_i}$. ThereforeConversely, every displayed matrix commutes with $R(g)$, proving both necessity and sufficiency. This commutant of a semisimple representation has complex dimension three. The parameters may vanish: $P$ was not required to be invertible.