Contragredient representation

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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.