= Inverse-adjoint representation
{title2=$D^\sharp(g)=D(g^{-1})^\dagger$}
For a complex finite-dimensional <group representation>, the displayed matrices give another <group representation>: taking inverses and taking the <conjugate transpose> each reverse multiplication order, so their composition preserves it. This operation is involutive. With the standard <Hermitian inner product>, an invariant <vector subspace> $W$ for $D^\sharp$ has <orthogonal complement> invariant for $D$, because $D(g)^\dagger W=D^\sharp(g^{-1})W\subseteq W$. Thus $\langle w,D(g)v\rangle=\langle D(g)^\dagger w,v\rangle=0$ for $v\in W^\perp,w\in W$. It follows that $D^\sharp$ is an <irreducible representation> exactly when $D$ is. No finiteness assumption on the <group> or unitarity assumption on the <group representation> is needed.
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