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
In the case of a field of characteristic zero closed under 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 .