In Fulton and Harris' Representation Theory, right at the beginning when they introduce representations, they note
The dual $V^{\ast} = \mbox{Hom}(V,{\mathbb C})$ of $V$ is also a representation, though not in the most obvious way: we want the rwo representations of $G$ to respect the natural pairing (denoted $\langle\hspace{.1in}, \hspace{.1in}\rangle$) between $V^{\ast}$ and $V$, so that if $\rho:G\rightarrow \mbox{GL}(V)$ is a representation and $\rho^{\ast}:G\rightarrow \mbox{GL}(V^{\ast})$ is the dual, we should have $\langle\rho^{\ast}(g)(v^{\ast}), \rho(g)(v)\rangle = \langle v^{\ast},v\rangle$ for all $g\in G$, $v\in V$, and $v^{\ast}\in V^{\ast}$. This in turn forces us to define the dual representation by $\rho^{\ast}(g) = ^{t}\rho(g^{-1}):V^{\ast}\rightarrow V^{\ast}$ for all $g\in G$.
I have a few questions about this.
- What is this natural pairing they are referring to? Is it that we can make our basis such that $e^{\ast}_{i}(e_{j}) = \delta_{ij}$?
- (This may be answered by the question above) What is the equality between these two relationships implying?
- What is this notation in the definition of the dual representation -- is this the transpose of the image of the inverse of $g$? Where is this coming from?