We introduced the dual representation of a representation of a finite group in the third construction in this post. We will keep notations we used in that post. I’m now going to show that is indeed a representation and it is irreducible if and only if is irreducible.

**Remark 1**. is a representation of

*Proof*. First we need to show that is well-defined, i.e. for all The fact that is -linear follows trivially from the fact that the elements of are linear. Also, if for some and then for all But, since is an isomorphism, is onto and hence Finally, is a group homomorphism because if and then

**Remark 2**. Suppose that is a -submodule of and let

1) is a -submodule of

2) If is irreducible, then is irreducible.

*Proof*. 1) over for all because So

2) Since is irreducible, either or Clearly if then Now, supppose that Let and choose a basis for Consider the natural linear map which is injective because But because are all -subspaces of Hence

which implies that and so

**Remark 3**. Suppose that is a -submodule of and let

1) is a -submodule of

2) If is irreducible, then is irreducible.

*Proof*. 1) Since we have for all

2) Since is irreducible, either or Clearly if then So suppose that If then there exists a -subspace of such that Choose and extend it to a basis for Now define by Clearly Contradiction!

how can i prove this when G is lie group ??differentiable of f:v_______c?

How do you know that V/ ker f_i is a subspace of C?

Because and so is a -linear map. Hence is isomorphic to a subspace of .