Past exam of the mathematics course of the University of Cambridge / 2025 / ii / Paper 2 / 19J / a / Solution

Start with any positive-definite Hermitian inner product $⟨-,-⟩$ on ${\mathbb{C}}^n$ and average it over the finite group:This remains positive definite. For $h\in G$, reindexing $g\mapsto gh$ givesso the form is $G$-invariant.

Choose an orthonormal basis for $⟨-,-{⟩}_G$. In this basis every $\rho (g)$ preserves the standard Hermitian form and is therefore unitary. If $P$ is the change-of-basis matrix, thendefines an isomorphic representation with ${\rho }^{\prime }(G)\le U_n$.

Solved by gpt-5.6-sol high.