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Dual (contragredient) representation

The dual representation acts on linear functionals by precomposition with the inverse group action; infinitesimally, it is the negative transpose.

Let VV be a finite-dimensional vector space. If ρ:G→GL(V)\rho:G\to \mathrm{GL}(V) is a GG, its dual, or contragredient, representation on V∗V^* is

ρ∗:G→GL(V∗),(ρ∗(g)λ)(v):=λ(ρ(g−1)v).\rho^*:G\to \mathrm{GL}(V^*),\qquad (\rho^*(g)\lambda)(v) := \lambda\bigl(\rho(g^{-1})v\bigr).

In a chosen basis, ρ∗(g)\rho^*(g) is represented by (ρ(g−1))T(\rho(g^{-1}))^{\mathsf T}.

Lie algebra version

If π:g→gl(V)\pi:\mathfrak g\to \mathfrak{gl}(V) is a g\mathfrak g, its dual representation π∗:g→gl(V∗)\pi^*:\mathfrak g\to\mathfrak{gl}(V^*) is defined by

(π∗(X)λ)(v):=− λ(π(X)v).(\pi^*(X)\lambda)(v) := -\,\lambda\bigl(\pi(X)v\bigr).

In matrix form, π∗(X)=−π(X)T\pi^*(X)=-\pi(X)^{\mathsf T}.

These definitions are compatible under differentiation: if π=dρe\pi=d\rho_e, then d(ρ∗)e=π∗d(\rho^*)_e=\pi^*.

Remarks

Duals interact naturally with other constructions such as . In highest-weight theory, dualizing typically negates weights (compare and ).