Can someone please explain what is Dual representation in plain English. I read its definition on wikipedia and at many other places but could not develop an intution for it. Please explain in plain english without lots of mathematical definitions.
How is following a Dual Representation :
There are two ways of looking at this. The first is in coordinates, where you regard the representation as mapping your group to a group of matrices with coordinates in your favorite field. You can produce a new representation from this by taking each matrix in the original representation and inverting, then transposing, it.
The second is abstractly. A linear representation of a group is a group action on a vector space. The dual space is the set of linear functionals on the vector space. The only obvious way to define a representation on the dual space is by taking a given functional and precomposing it with the inverted representation on the vector space.
It may be worth asking why you invert the representation before precomposing with a functional (alternatively, why you invert the matrix before transposing it). There's a technical reason to do this: so that the dual representation actually turns into a group homomorphism. If $\rho:G\to GL(V)$, $g,h\in G$, $v\in V$, and $f\in V^*$, then $$\rho^*(gh)(f)(v) = f(\rho(gh)v) = f(\rho(h)\rho(g)v) = \rho^*(h)f(\rho(g)v)=\rho^*(h)\rho^*(g)f(v)$$ doesn't actually work.