Tensor decomposition, Sym representations and irreps.

[knowwhatyoudontknow]

TL;DR

Tensor decomposition, Sym[SUB]n[/SUB] representations and irreps.

New to group theory. I have 3 questions:

1. Tensor decomposition into T_ab = T_[ab] +T^(traceless)_{ab} + Tr(T_{ab}) leads to invariant subspaces. Is this enough to imply these subreps are irreducible?

2. The Symⁿ representations of a group are irreps. Why?

3. What is the connection between Symⁿ representations and tensor decomposition?

 

[martinbn]

You need to give more details and context. Are you looking at representations of finite groups?

1. I am not sure what the question is.

2. This doesn't seem right. A finite group has only finitely many irreducible representations. So the $Sym^n$ cannot be all irreducible.

3. Also not sure what you are asking.