Conjugate contragredient representation ($D^{\#}(g)=D(g^{-1}{)}^{†}$)

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 $⟨w,D(g)v⟩=⟨D^{\#}(g^{-1})w,v⟩=0$ 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.