I Tensor decomposition, Sym representations and irreps.

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Tensor decomposition, Sym[SUB]n[/SUB] representations and irreps.
New to group theory. I have 3 questions:

1. Tensor decomposition into Tab = 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?
 
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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.
 

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