Tensor decomposition, Sym representations and irreps.

In summary, the conversation discusses the connection between tensor decomposition and symmetric representations in group theory. The first question asks if tensor decomposition into specific subspaces implies irreducibility, while the second question inquires about the irreducibility of Symn representations. The third question seeks to understand the relationship between Symn representations and tensor decomposition. It is unclear if this discussion is about representations of finite groups.
  • #1
knowwhatyoudontknow
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TL;DR Summary
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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  • #2
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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