✎ Dual Representation [The Physics Travel Guide]

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====== Dual Representation ======

<tabbox Intuitive> 
For any representation, there is a dual representation, which acts on the dual vector space instead of the original vector space. The dual vector space consists of covectors, which are linear functions from the vectors to scalars.

In some cases, the dual representation and the original representation are the same (e.g. for representations with orthogonal matrices).

For unitary representations, the dual representation is the same as the complex-conjugate representation.

<tabbox Concrete> 
**Example**

The diagram below shows the defining representation of $SU(2)$ in its upper branch. To construct the dual representation, we let it act on the dual vector space, that is, the space of covectors. In this example, the dual representation is the same as the complex-conjugate representation ($\tilde{U}=U^*$). Moreover, the defining and complex-conjugate representations of $SU(2)$ are equivalent, that is, there is a similarity transformation that takes one to the other.

[{{ :advanced_tools:group_theory:representation_theory:su2_dual.jpg?nolink }}]

For a more detailed explanation of this diagram see [[https://esackinger.wordpress.com/|Fun with Symmetry]]. 
 
<tabbox Abstract>

<note tip>
The motto in this section is: //the higher the level of abstraction, the better//.
</note>

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