# Contragredient representation

## Definition

### 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 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 $\rho$.