Contragredient representation

Conceptual definition
Suppose $$\rho:G \to GL(V)$$ is a linear representation of a group $$G$$ on a vector space $$V$$ over a field $$K$$. The contragredient representation of $$\rho$$, sometimes denoted $$\rho^*$$, is a linear representation $$G \to GL(V^*)$$ on the dual space $$V^*$$ (i.e., the space of linear functionals on $$V$$) as follows. For $$f \in V^*$$, i.e., $$f:V \to K$$ linear, we define $$\rho^*(g)(f)$$ as the element of $$V^*$$ given by $$v \mapsto f(g^{-1}(v))$$.

Definition in matrix terms
Suppose $$\rho:G \to GL(n,K)$$ is a finite-dimensional linear representation of a group $$G$$ over a field $$K$$. The contragredient representation $$\rho^*: G \to GL(n,K)$$ is defined as the composite of the defining ingredient::transpose-inverse map with $$\rho$$, i.e.:

$$\rho^*(g) = ((\rho(g))^T)^{-1}$$

In the case of a field of characteristic zero closed under complex conjugation
Suppose $$K$$ is a subfield of the complex numbers that is closed under complex conjugation. Then, the contragredient representation to any representation over $$K$$ of a finite group $$G$$ is equivalent to the linear representation obtained by composing complex conjugation with $$\rho$$.