Dual representation

In mathematics, if G is a group and ρ is a linear representation of it on the vector space V, then the dual representation ρ* is defined over the dual vector space V* as follows:^[1]

ρ*(g) is the transpose of ρ(g⁻¹), that is, ρ*(g) = ρ(g⁻¹)^T for all g ∈ G.

The dual representation is also known as the contragredient representation.

If g is a Lie algebra and π is a representation of it on the vector space V, then the dual representation π* is defined over the dual vector space V* as follows:^[2]

π*(X) = −π(X)^T for all X ∈ g.

In both cases, the dual representation is a representation in the usual sense.

Motivation

In representation theory, both vectors in V and linear functionals in V* are considered as column vectors so that the representation can act (by matrix multiplication) from the left. Given a basis for V and the dual basis for V*, the action of a linear functional φ on v, φ(v) can be expressed by matrix multiplication,

$\langle\varphi, v\rangle \equiv \varphi(v) = \varphi^Tv$,

where the superscript T is matrix transpose. Consistency requires

$\langle{\rho}^*(g)\varphi, \rho(g)v\rangle = \langle\varphi, v\rangle.$^[3]

With the definition given,

$\langle{\rho}^*(g)\varphi, \rho(g)v\rangle = \langle\rho(g^{-1})^T\varphi, \rho(g)v\rangle = (\rho(g^{-1})^T\varphi)^T \rho(g)v = \varphi^T\rho(g^{-1})\rho(g)v = \varphi^Tv = \langle\varphi, v\rangle$.

For the Lie algebra representation one chooses consistency with a possible group representation. Generally, if Π is a representation of a Lie group, then π given by

$\pi(X) = \frac{d}{dt}\Pi(e^{tX})|_{t = 0}.$

is a representation of its Lie algebra. If Π* is dual to Π, then its corresponding Lie algebra representation π* is given by

$\pi^*(X) = \frac{d}{dt}\Pi^*(e^{tX})|_{t = 0} = \frac{d}{dt}\Pi(e^{-tX})^T|_{t = 0} = -\pi(X)^T.$.^[4]

Generalization

A general ring module does not admit a dual representation. Modules of Hopf algebras do, however.

See also

• Complex conjugate representation
• Kirillov Character Formula

References

1. Lecture 1 of Fulton, William; Harris, Joe (1991), Representation theory. A first course, Graduate Texts in Mathematics, Readings in Mathematics 129, New York: Springer-Verlag, ISBN 978-0-387-97495-8, MR 1153249, ISBN 978-0-387-97527-6
2. Lecture 8 of Fulton, William; Harris, Joe (1991), Representation theory. A first course, Graduate Texts in Mathematics, Readings in Mathematics 129, New York: Springer-Verlag, ISBN 978-0-387-97495-8, MR 1153249, ISBN 978-0-387-97527-6
3. Lecture 1, page 4 of Fulton, William; Harris, Joe (1991), Representation theory. A first course, Graduate Texts in Mathematics, Readings in Mathematics 129, New York: Springer-Verlag, ISBN 978-0-387-97495-8, MR 1153249, ISBN 978-0-387-97527-6
4. Lecture 8, page 111 of Fulton, William; Harris, Joe (1991), Representation theory. A first course, Graduate Texts in Mathematics, Readings in Mathematics 129, New York: Springer-Verlag, ISBN 978-0-387-97495-8, MR 1153249, ISBN 978-0-387-97527-6