I have a doubt concerning the dual representation. Can someone check that what I wrote is correct please?
Let $A: V \to V$ be linear, the dual map $A^T : V^* \to V^*$ is defined by
$$ A^*(f)(x)=f(A(x))$$
or in other words,
$$ \langle{A^Tf,x}\rangle=\langle{f,Ax}\rangle $$
Let $e_i$ be a basis for $V$ and $e^i$ its dual basis
$$ \langle{A^T e^i,e_j}\rangle=\langle{e^i,Ae_j}\rangle=A_{ji}$$
We get the transposed matrix.
Now let $\rho: G \to \text{GL}(V) $ be a representation. The dual representation
$$ \rho: G \to \text{GL}(V^*) $$
is given by
$$ \rho^*(g)=\rho(g^{-1})^T $$
where the inverse insures that one gets a homomorphism. Relatively to the same bases, one gets
\begin{align*} \langle{\rho^*(g) e^i,e_j}\rangle &=\langle{\rho^T(g^{-1}) e^i,e_j}\rangle \\ &=\langle{e^i,\rho(g^{-1}) e_j}\rangle\\ &=\rho(g)^{-1}_{ji} \end{align*} the inverse of the transpose.
Is this correct?