= Conjugate contragredient representation
{title2=$D^\#(g)=D(g^{-1})^\dagger$}
For a finite-dimensional complex <group representation>, combining inverse and <conjugate transpose> gives another representation: both operations reverse multiplication order. This is the complex conjugate of the algebraic <dual representation> in matrices. If $W$ is invariant under $D^\#$, its orthogonal complement is invariant under $D$, because $\langle w,D(g)v\rangle=\langle D^\#(g^{-1})w,v\rangle=0$ for $w\in W$, $v\in W^\perp$. Thus irreducibility is preserved. Applying the construction twice returns $D$, so the equivalence of irreducibility works in both directions for arbitrary groups in finite dimension.
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