# Dual representation

A dual representation (also called the contragredient representation) is the representation that a group or [Lie algebra](https://www.edgepedia.org/lie-algebra) induces on the dual space of its representation space: if \( \phi \) is a representation of a group \( G \) on a vector space \( V \), the dual representation \( \phi^{*} \) acts on the dual space \( V^{*} \) by \( \phi^{*}(g) = \phi(g^{-1})^{*} \), where \( * \) denotes taking adjoints<sup>[1](https://encyclopediaofmath.org/wiki/Contragredient_representation)</sup>. The inverse and the adjoint together ensure that the natural pairing between \( V \) and \( V^{*} \) is preserved, which is the defining property of the construction.

| Key fact | Detail |
|---|---|
| Definition | \( \phi^{*}(g) = \phi(g^{-1})^{*} \) on \( V^{*} \), for a representation \( \phi \) of \( G \) on \( V \)<sup>[1](https://encyclopediaofmath.org/wiki/Contragredient_representation)</sup> |
| Matrix form | In the dual basis, the matrix of \( \phi^{*}(g) \) is the transpose of the inverse of the matrix of \( \phi(g) \)<sup>[1](https://encyclopediaofmath.org/wiki/Contragredient_representation)</sup> |
| Lie algebra version | \( \rho_{V^{*}}(x) = -\rho_{V}(x)^{*} \); equivalently \( (d\phi(X)x,\, y) = -(x,\, d\psi(X)y) \) for contragredient pairs<sup>[2](https://math.mit.edu/~etingof/reprbook.pdf)</sup><sup> • </sup><sup>[1](https://encyclopediaofmath.org/wiki/Contragredient_representation)</sup> |
| Irreducibility | If \( \phi \) is irreducible, then so is \( \phi^{*} \)<sup>[1](https://encyclopediaofmath.org/wiki/Contragredient_representation)</sup> |
| Weights | The weights of \( \phi^{*} \) are the negatives of those of \( \phi \); the lowest weight of \( \phi^{*} \) is opposite the highest weight of \( \phi \)<sup>[1](https://encyclopediaofmath.org/wiki/Contragredient_representation)</sup> |
| Self-duality criterion | For finite-dimensional representations, \( \phi \cong \phi^{*} \) exactly when \( V \) carries a non-degenerate \( \phi(G) \)-invariant bilinear form; for irreducible \( \phi \), every nonzero invariant bilinear form is non-degenerate and, over an algebraically closed field, is symmetric or skew-symmetric<sup>[1](https://encyclopediaofmath.org/wiki/Contragredient_representation)</sup><sup> • </sup><sup>[11](https://math.berkeley.edu/~serganov/math252/Bookrep.pdf)</sup> |
| Special case | The contragredient of the adjoint representation is the coadjoint representation, on \( \mathfrak{g}^{*} \)<sup>[3](https://encyclopediaofmath.org/index.php?title=Adjoint_action)</sup> |

## Definition and matrix form

The dual space \( V^{*} \) consists of the linear functionals on \( V \). Given a representation \( \phi : G \to \mathrm{GL}(V) \), the formula \( \phi^{*}(g) = \phi(g^{-1})^{*} \) defines a representation of the same group \( G \) on \( V^{*} \)<sup>[1](https://encyclopediaofmath.org/wiki/Contragredient_representation)</sup>. The inverse appears because a functional \( \ell \in V^{*} \) must act on vectors that have already been transformed: \( (g \cdot \ell)(v) = \ell(g^{-1} \cdot v) \)<sup>[4](https://courses.washington.edu/mnotes/liegroups/notes.pdf)</sup>. The two inverses (one from \( g^{-1} \), one from the adjoint) cancel in the group law, so \( \phi^{*} \) is a genuine homomorphism<sup>[4](https://courses.washington.edu/mnotes/liegroups/notes.pdf)</sup>.

In coordinates the construction is explicit. If \( V \) is finite-dimensional with basis \( (e) \) and \( (f) \) is the dual basis of \( V^{*} \), then the matrix of \( \phi^{*}(g) \) in the basis \( (f) \) is obtained from the matrix of \( \phi(g) \) by taking the transpose of the inverse<sup>[1](https://encyclopediaofmath.org/wiki/Contragredient_representation)</sup>. Written in matrices, the dual representation is often called the contragredient representation<sup>[4](https://courses.washington.edu/mnotes/liegroups/notes.pdf)</sup>.

The definition generalizes. If \( W \) is a vector space over the same field \( k \) and \( (\ ,\ ) \) is a non-degenerate bilinear form (a pairing) on \( V \times W \), a representation \( \psi \) of \( G \) on \( W \) is contragredient to \( \phi \) with respect to this form when

\[ (\phi(g)x,\ y) = (x,\ \psi(g^{-1})y) \]

for all \( g \in G \), \( x \in V \), \( y \in W \)<sup>[1](https://encyclopediaofmath.org/wiki/Contragredient_representation)</sup>. Taking \( W = V^{*} \) with the natural pairing recovers the basic case. A concrete instance: for \( G \) the general linear group of a finite-dimensional space \( V \), the natural representation on covariant tensors of a fixed rank is contragredient to the natural representation on contravariant tensors of the same rank<sup>[1](https://encyclopediaofmath.org/wiki/Contragredient_representation)</sup>.

## Lie algebra version

For a Lie algebra \( \mathfrak{g} \), the dual representation on \( V^{*} \) is defined by \( \rho_{V^{*}}(x) = -\rho_{V}(x)^{*} \)<sup>[2](https://math.mit.edu/~etingof/reprbook.pdf)</sup>. The minus sign is forced by the requirement that the assignment preserve the Lie bracket: the transpose reverses the order of multiplication, and the negative sign compensates<sup>[5](https://androma.org/theorems/9391)</sup>.

Equivalently, if \( \phi \) and \( \psi \) are representations of a [Lie group](https://www.edgepedia.org/lie-group) \( G \) with Lie algebra \( \mathfrak{g} \) on spaces \( V \) and \( W \) contragredient with respect to a pairing \( (\ ,\ ) \), their differentials satisfy

\[ (d\phi(X)x,\ y) = -(x,\ d\psi(X)y) \]

for all \( X \in \mathfrak{g} \), \( x \in V \), \( y \in W \). Lie algebra representations satisfying this condition are themselves called contragredient with respect to the pairing<sup>[1](https://encyclopediaofmath.org/wiki/Contragredient_representation)</sup>.

For an associative algebra \( A \), the same idea produces the dual representation \( T^{\star} : A^{\mathrm{op}} \to \mathrm{End}_{k}(E^{\star}) \) of the opposite algebra, defined by \( T^{\star}(a)(\varphi)(e) = \varphi(T(a)(e)) \); the opposite algebra appears because the transpose reverses multiplication<sup>[6](https://encyclopediaofmath.org/wiki/Representation_of_an_associative_algebra)</sup>.

## Weights and the Dynkin diagram

For a complex, connected, simply connected semisimple Lie group \( G \) and an irreducible finite-dimensional representation \( \phi \), the weights of \( \phi^{*} \) are the negatives of the weights of \( \phi \), with the same multiplicities; the lowest weight of \( \phi^{*} \) is opposite to the highest weight of \( \phi \)<sup>[1](https://encyclopediaofmath.org/wiki/Contragredient_representation)</sup><sup> • </sup><sup>[5](https://androma.org/theorems/9391)</sup>.

The highest weight of \( \phi^{*} \) is obtained from that of \( \phi \) by an automorphism \( \nu \) of the [Dynkin diagram](https://www.edgepedia.org/dynkin-diagram) of simple roots. This automorphism preserves each connected component; on a component of type \( A_{r} \), \( D_{2r+1} \), or \( E_{6} \) it is the unique element of order 2 in the diagram's automorphism group, and on all other components it is the identity<sup>[1](https://encyclopediaofmath.org/wiki/Contragredient_representation)</sup>. This is why dual representations of, say, symplectic groups are isomorphic to their originals while certain representations of \( \mathrm{SL}_{n} \) are not.

## Self-duality

A representation is self-dual when \( \phi \) and \( \phi^{*} \) are equivalent. For finite-dimensional representations, the criterion is bilinear: \( \phi \) and \( \phi^{*} \) are equivalent if and only if there is a non-degenerate bilinear form on \( V \) invariant under \( \phi(G) \); for an irreducible representation, every nonzero invariant bilinear form is non-degenerate, and over an algebraically closed field it is either symmetric or skew-symmetric<sup>[1](https://encyclopediaofmath.org/wiki/Contragredient_representation)</sup><sup> • </sup><sup>[11](https://math.berkeley.edu/~serganov/math252/Bookrep.pdf)</sup>.

The standard representation of \( \mathrm{SU}(2) \) illustrates the skew-symmetric case. On \( V = \mathbb{C}^{2} \), the form \( q(v, w) = v_{1}w_{2} - w_{1}v_{2} \) is non-degenerate, skew-symmetric, and preserved because \( \det A = 1 \) for \( A \in \mathrm{SU}(2) \); it therefore identifies \( V \) with \( V^{*} \) as representations<sup>[4](https://courses.washington.edu/mnotes/liegroups/notes.pdf)</sup>.

## The coadjoint representation

Applying the construction to the adjoint representation \( \mathrm{Ad} : G \to \mathrm{GL}(\mathfrak{g}) \) gives the coadjoint representation \( \mathrm{Ad}^{*} : G \to \mathrm{GL}(\mathfrak{g}^{*}) \) on the dual of the Lie algebra<sup>[3](https://encyclopediaofmath.org/index.php?title=Adjoint_action)</sup>. Coadjoint orbits \( \mathcal{O}_{\Lambda} = \{\mathrm{Ad}^{*}_{u}\Lambda \mid u \in G\} \) carry symplectic structures and serve as phase spaces in Hamiltonian mechanics; the Lie-Poisson structure lives on \( \mathfrak{g}^{*} \) itself<sup>[7](https://authors.library.caltech.edu/records/rpq2q-xr717)</sup>. In conformal field theory, fields lying in the coadjoint representation of infinite-dimensional algebras such as Virasoro and affine Lie algebras have short-distance operator product expansions computed from the adjoint representation and its dual<sup>[8](https://arxiv.org/abs/hep-th/0211021)</sup>.

## Contragredient representations of p-adic groups

For smooth representations of p-adic groups the construction needs a modification. The linear dual \( V' = \mathrm{Hom}_{\mathbb{C}}(V, \mathbb{C}) \) of a smooth representation \( (\pi, V) \) carries the action \( [\pi'(g)\lambda](v) = \lambda(\pi(g^{-1})v) \), but this action is rarely smooth. The contragredient \( (\tilde{\pi}, \tilde{V}) \) is defined on \( \tilde{V} = (V')_{\infty} \), the space of smooth vectors in the linear dual<sup>[9](https://math.stanford.edu/~conrad/modseminar/pdf/L09.pdf)</sup>.

The contragredient defines a contravariant functor on the category of smooth representations, but in general it is not a duality: the natural map from \( V \) to its double contragredient exists but need not be an isomorphism. It becomes a duality on the subcategory of admissible representations, those for which the fixed subspace \( V^{H} \) is finite-dimensional for every open subgroup \( H \)<sup>[9](https://math.stanford.edu/~conrad/modseminar/pdf/L09.pdf)</sup>. For irreducible admissible representations of classical groups over non-archimedean local fields, the contragredient of \( \pi \) is isomorphic to a twist of \( \pi \) by an automorphism of \( G \)<sup>[10](https://arxiv.org/abs/0903.1418)</sup>.

## References

1. [Contragredient representation, Encyclopedia of Mathematics](https://encyclopediaofmath.org/wiki/Contragredient_representation)
2. [Pavel Etingof, Oleg Golberg, Sebastian Hensel, Tiankai Liu, Alex Schwendner, Dmitry Vaintrob, Elena Yudovina. Introduction to representation theory, §2.14](https://math.mit.edu/~etingof/reprbook.pdf)
3. [Adjoint action, Encyclopedia of Mathematics](https://encyclopediaofmath.org/index.php?title=Adjoint_action)
4. [Lie Groups and Representation Theory, lecture notes, University of Washington](https://courses.washington.edu/mnotes/liegroups/notes.pdf)
5. [Highest Weight of the Dual Irreducible Module, Androma](https://androma.org/theorems/9391)
6. [Representation of an associative algebra, Encyclopedia of Mathematics](https://encyclopediaofmath.org/wiki/Representation_of_an_associative_algebra)
7. [Marsden, Ratiu, Schmid, Spencer, Weinstein (1983). Hamiltonian systems with symmetry, coadjoint orbits and plasma physics, CaltechAUTHORS](https://authors.library.caltech.edu/records/rpq2q-xr717)
8. [Gates, Linch, Phillips, Rodgers (2003). Short Distance Expansion from the Dual Representation of Infinite Dimensional Lie Algebras, arXiv:hep-th/0211021](https://arxiv.org/abs/hep-th/0211021)
9. [Representations of p-adic groups, lecture notes, Stanford University](https://math.stanford.edu/~conrad/modseminar/pdf/L09.pdf)
10. [Dual pairs and contragredients of irreducible representations, arXiv:0903.1418](https://arxiv.org/abs/0903.1418)
11. [math.berkeley.edu](https://math.berkeley.edu/~serganov/math252/Bookrep.pdf)

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*Topic: Encyclopedia › Physical world and mathematics › Mathematics and statistics › Numbers and algebra › Advanced algebraic structures › Lie theory › Lie representations and modules › Overview of Lie algebra representations*

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