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

Further information: contragredient representation of finite group equals composite with 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 $\rho$.