Given a representation $R:G\to GL(V)$
, the

The dual vector space ${V}^{*}$ is the space $\mathrm{hom}(V,\mathbf{C})$ of linear maps from $V$ to $\mathbf{C}$ . Equivalently, if we think of $V$ as a space of column vectors by picking a basis of $V$ then we can think of ${V}^{*}$ as the space of row vectors: given an element $\alpha \in \mathrm{hom}(V,\mathbf{C})$ , we get a row vector $\underset{\xaf}{\alpha}$ as follows. Pick a basis of $V$ so that elements can be written as column vectors $v=\left(\begin{array}{c}\hfill {v}_{1}\hfill \\ \hfill \mathrm{\vdots}\hfill \\ \hfill {v}_{n}\hfill \end{array}\right)$ . Then $\alpha (v)={\alpha}_{1}{v}_{1}+\mathrm{\cdots}+{\alpha}_{n}{v}_{n}$ for some coefficients ${\alpha}_{i}$ . Define $\underset{\xaf}{\alpha}=\left(\begin{array}{ccc}\hfill {\alpha}_{1}\hfill & \hfill \mathrm{\cdots}\hfill & \hfill {\alpha}_{n}\hfill \end{array}\right)$ (so $\alpha (v)=\underset{\xaf}{\alpha}\cdot v$ ).
Conversely, if you have a row vector $w=\left(\begin{array}{ccc}\hfill {w}_{1}\hfill & \hfill \mathrm{\cdots}\hfill & \hfill {w}_{n}\hfill \end{array}\right)$ then we get the linear map $\overline{w}\in \mathrm{hom}(V,\mathbf{C})$ , defined by $\overline{w}(v)={w}_{1}{v}_{1}+\mathrm{\cdots}+{w}_{n}{v}_{n}$ .

The dual representation is defined as follows. For each $g\in G$ , define ${R}^{*}(g):{V}^{*}\to {V}^{*}$ to be the map which sends a row vector $\underset{\xaf}{\alpha}$ to $\underset{\xaf}{\alpha}R{(g)}^{1}$ . In other words, $$\underset{\xaf}{{R}^{*}(g)\alpha}=\underset{\xaf}{\alpha}R{(g)}^{1}.$$
The messing with underlines is just to distinguish between applying a linear map and multiplying a matrix, because we want to multiply on the right.