Intuitive

For any representation, we can construct a dual representation. To do that we let the representation act on the dual vector space instead of the original vector space. The dual vector space contains covectors, that is, linear functions that map vectors to scalars.

In some cases, the dual representation is equivalent to the original one (e.g. for representations with orthogonal matrices).

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

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.

For a more detailed explanation of this diagram see Fun with Symmetry.

Abstract

Why is it interesting?