# Contragredient representation

## Contents

## Definition

### Conceptual definition

Suppose is a linear representation of a group on a vector space over a field . The **contragredient representation** of , sometimes denoted , is a linear representation on the dual space (i.e., the space of linear functionals on ) as follows. For , i.e., linear, we define as the element of given by .

### Definition in matrix terms

Suppose is a finite-dimensional linear representation of a group over a field . The contragredient representation is defined as the composite of the transpose-inverse map with , i.e.:

### 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 is a subfield of the complex numbers that is closed under complex conjugation. Then, the contragredient representation to any representation over of a finite group is equivalent to the linear representation obtained by composing complex conjugation with .