Editing Dual representation

{{Use American English|date = March 2019}}
{{Short description|Group representation}}
{{about|a mathematical concept|the psychological concept|Dual representation (psychology)|other uses}}
In [[mathematics]], if {{math|''G''}} is a [[group (mathematics)|group]] and {{math|ρ}} is a [[linear representation]] of it on the [[vector space]] {{math|''V''}}, then the '''dual representation''' {{math|ρ*}} is defined over the [[dual vector space]] {{math|''V''*}} as follows:<ref>Lecture 1 of {{Fulton-Harris}}</ref><ref>{{harvnb|Hall|2015}} Section 4.3.3</ref>

:{{math|ρ*(''g'')}} is the [[transpose of a linear map|transpose]] of {{math|ρ(''g''<sup>−1</sup>)}}, that is, {{math|ρ*(''g'')}} = {{math|ρ(''g''<sup>−1</sup>)<sup>T</sup>}} for all {{math|''g'' ∈ ''G''}}.

The dual representation is also known as the '''contragredient representation'''.

If {{math|'''g'''}} is a [[Lie algebra]] and {{math|π}} is a representation of it on the vector space {{math|''V''}}, then the dual representation {{math|π*}} is defined over the dual vector space {{math|''V''*}} as follows:<ref>Lecture 8 of {{Fulton-Harris}}</ref>

:{{math|π*(''X'')}} = {{math|−π(''X'')<sup>''T''</sup>}} for all {{math|''X'' ∈ '''g'''}}.

The motivation for this definition is that Lie algebra representation associated to the dual of a [[Lie group]] representation is computed by the above formula. But the definition of the dual of a Lie algebra representation makes sense even if it does not come from a Lie group representation.

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

==Properties==
===Irreducibility and second dual===
If a (finite-dimensional) representation is irreducible, then the dual representation is also irreducible<ref>{{harvnb|Hall|2015}} Exercise 6 of Chapter 4</ref>—but not necessarily isomorphic to the original representation. On the other hand, the dual of the dual of any representation is isomorphic to the original representation.

===Unitary representations===
Consider a ''unitary'' representation <math>\rho</math> of a group <math>G</math>, and let us work in an orthonormal basis. Thus, <math>\rho</math> maps <math>G</math> into the group of unitary matrices. Then the abstract transpose in the definition of the dual representation may be identified with the ordinary matrix transpose. Since the adjoint of a matrix is the complex conjugate of the transpose, the transpose is the conjugate of the adjoint. Thus, <math>\rho^\ast(g)</math> is the complex conjugate of the adjoint of the inverse of <math>\rho(g)</math>. But since <math>\rho(g)</math> is assumed to be unitary, the adjoint of the inverse of <math>\rho(g)</math> is just <math>\rho(g)</math>.

The upshot of this discussion is that when working with unitary representations in an orthonormal basis, <math>\rho^*(g)</math> is just the complex conjugate of <math>\rho(g)</math>.

===The SU(2) and SU(3) cases===
In the representation theory of SU(2), the dual of each irreducible representation does turn out to be isomorphic to the representation. But for the [[Lie_algebra_representation#The_case_of_sl(3,C)|representations of SU(3)]], the dual of the irreducible representation with label <math>(m_1,m_2)</math> is the irreducible representation with label <math>(m_2,m_1)</math>.<ref>{{harvnb|Hall|2015}} Exercise 3 of Chapter 6</ref> In particular, the standard three-dimensional representation of SU(3) (with highest weight <math>(1,0)</math>) is not isomorphic to its dual. In the [[Quark_model#Mesons|theory of quarks]] in the physics literature, the standard representation and its dual are called "<math>3</math>" and "<math>\bar 3</math>."
[[File:Dual_representations_of_SU(3).png|thumb|right|Two nonisomorphic dual representations of SU(3), with highest weights (1,2) and (2,1)]]

===General semisimple Lie algebras===
More generally, in the [[Lie_algebra_representation#Classifying_finite-dimensional_representations_of_Lie_algebras|representation theory of semisimple Lie algebras]] (or the closely related [[Compact_group#Representation_theory_of_a_connected_compact_Lie_group|representation theory of compact Lie groups]]), the weights of the dual representation are the ''negatives'' of the weights of the original representation.<ref>{{harvnb|Hall|2015}} Exercise 10 of Chapter 10</ref> (See the figure.) Now, for a given Lie algebra, if it should happen that operator <math>-I</math> is an element of the [[Semisimple_Lie_algebra#Weyl_group|Weyl group]], then the weights of every representation are automatically invariant under the map <math>\mu\mapsto -\mu</math>. For such Lie algebras, ''every'' irreducible representation will be isomorphic to its dual. (This is the situation for SU(2), where the Weyl group is <math>\{I,-I\}</math>.) Lie algebras with this property include the odd orthogonal Lie algebras <math>\operatorname{so}(2n+1;\mathbb C)</math> (type <math>B_n</math>) and the symplectic Lie algebras <math>\operatorname{sp}(n;\mathbb C)</math> (type <math>C_n</math>). 

If, for a given Lie algebra, <math>-I</math> is ''not'' in the Weyl group, then the dual of an irreducible representation will generically not be isomorphic to the original representation. To understand how this works, we note that there is always a [[Root_system#Weyl_chambers_and_the_Weyl_group|unique Weyl group element]] <math>w_0</math> mapping the negative of the fundamental Weyl chamber to the fundamental Weyl chamber. Then if we have an irreducible representation with highest weight <math>\mu</math>, the ''lowest'' weight of the dual representation will be <math>-\mu</math>. It then follows that the ''highest'' weight of the dual representation will be <math>w_0\cdot(-\mu)\,</math>.<ref>{{harvnb|Hall|2015}} Exercise 10 of Chapter 10</ref> Since we are assuming <math>-I</math> is not in the Weyl group, <math>w_0</math> cannot be <math>-I</math>, which means that the map <math>\mu\mapsto w_0\cdot(-\mu)</math> is not the identity. Of course, it may still happen that for certain special choices of <math>\mu</math>, we might have <math>\mu=w_0\cdot(-\mu)</math>. The adjoint representation, for example, is always isomorphic to its dual.

In the case of SU(3) (or its complexified Lie algebra, <math>\operatorname{sl}(3;\mathbb C)</math>), we may choose a base consisting of two roots <math>\{\alpha_1,\alpha_2\}</math> at an angle of 120 degrees, so that the third positive root is <math>\alpha_3=\alpha_1+\alpha_2</math>. In this case, the element <math>w_0</math> is the reflection about the line perpendicular to <math>\alpha_3</math>. Then the map <math>\mu\mapsto w_0\cdot(-\mu)</math> is the reflection about the line ''through'' <math>\alpha_3</math>.<ref>{{harvnb|Hall|2015}} Exercise 3 of Chapter 6</ref> The self-dual representations are then the ones that lie along the line through <math>\alpha_3</math>. These are the representations with labels of the form <math>(m,m)</math>, which are the representations whose weight diagrams are ''regular'' hexagons.

==Motivation==
In representation theory, both vectors in {{math|''V''}} and linear functionals in {{math|''V''*}} are considered as ''column vectors'' so that the representation can act (by matrix multiplication) from the ''left''. Given a basis for {{math|''V''}} and the dual basis for {{math|''V''*}}, the action of a linear functional {{math|φ}} on {{math|''v''}}, {{math|φ(v)}} can be expressed by matrix multiplication,
:<math>\langle\varphi, v\rangle \equiv \varphi(v) = \varphi^Tv</math>,
where the superscript {{math|''T''}} is matrix transpose. Consistency requires
:<math>\langle{\rho}^*(g)\varphi, \rho(g)v\rangle = \langle\varphi, v\rangle.</math><ref>Lecture 1, page 4 of {{Fulton-Harris}}</ref>
With the definition given,
:<math>\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.</math>

For the Lie algebra representation one chooses consistency with a possible group representation. Generally, if {{math|Π}} is a representation of a Lie group, then {{math|π}} given by
:<math>\pi(X) = \frac{d}{dt}\Pi(e^{tX})|_{t = 0}.</math>
is a representation of its Lie algebra. If {{math|Π*}} is dual to {{math|Π}}, then its corresponding Lie algebra representation {{math|π*}} is given by
:<math>\pi^*(X) = \frac{d}{dt}\Pi^*(e^{tX})|_{t = 0} = \frac{d}{dt}\Pi(e^{-tX})^T|_{t = 0} = -\pi(X)^T.</math>&nbsp;&nbsp;&nbsp;<ref>Lecture 8, page 111 of {{Fulton-Harris}}</ref>

==Example==
Consider the group <math>G=U(1)</math> of complex numbers of absolute value 1. The irreducible representations are all one dimensional, as a consequence of [[Schur's lemma]]. The irreducible representations are parameterized by integers <math>n</math> and given explicitly as
:<math>\rho_n(e^{i\theta})=[e^{in\theta}].</math>
The dual representation to <math>\rho_n</math> is then the inverse of the transpose of this one-by-one matrix, that is,
:<math>\rho_n^*(e^{i\theta})=[e^{-in\theta}]=\rho_{-n}(e^{i\theta}).</math>
That is to say, the dual of the representation <math>\rho_n</math> is <math>\rho_{-n}</math>.

==Generalization==

A general ring [[Module (mathematics)|module]] does not admit a dual representation. Modules of [[Hopf algebra]]s do, however.

==See also==
* [[Complex conjugate representation]]
*[[Tensor product of representations]]
* [[Kirillov Character Formula]]

==References==
* {{citation|first=Brian C.|last=Hall|title=Lie Groups, Lie Algebras, and Representations: An Elementary Introduction|edition= 2nd|series=Graduate Texts in Mathematics|volume=222 |publisher=Springer|year=2015|isbn=978-3319134666}}.
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[[Category:Representation theory of groups]]