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

An invariant bilinear form $B$ defines a linear mapInvariance givesso ${\Phi }_B$ is an $G$-homomorphism. Its kernel is a $G$-subrepresentation of irreducible $V$. Hence either ${\Phi }_B=0$, so $B=0$, or ${\Phi }_B$ is injective and therefore a linear isomorphism $V\to V^{\ast }$. In the latter case $B$ is a nondegenerate.

If $B$ and $C$ are two nonzero invariant forms, thenBy Schur lemma, this endomorphism is scalar, so $B=\lambda C$. Cases involving the zero form are immediate. Thus any two invariant bilinear forms are proportional.

Given a nonzero invariant form $B$, its transposeis again nonzero and $G$-invariant. By part (i), $B^T=\lambda B$ for some nonzero scalar $\lambda$. Transposing once more givesSince $B\ne 0$, ${\lambda }^2=1$, and hence $\lambda \in \{1,-1\}$. ThereforeThus every nonzero invariant bilinear form on an irreducible complex representation is either symmetric or alternating.