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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1. What is tensor decomposition?

Tensor decomposition is a mathematical method used to break down a higher-order tensor into a combination of lower-order tensors. It is commonly used in data analysis and machine learning to simplify complex data structures and extract meaningful information.

2. What are Sym representations?

Sym representations, also known as symmetric representations, are mathematical representations of a group or symmetry. They are used in the study of symmetry and group theory, and have applications in physics, chemistry, and other fields.

3. What are irreps?

Irreps, short for irreducible representations, are mathematical representations of a group or symmetry that cannot be broken down into smaller representations. They are important in the study of symmetry and group theory, and are used in various fields such as physics, chemistry, and mathematics.

4. How are tensor decomposition, Sym representations, and irreps related?

Tensor decomposition, Sym representations, and irreps are all related to the study of symmetry and group theory. Tensor decomposition can be used to break down a higher-order tensor into a combination of irreps, while Sym representations and irreps are both types of mathematical representations of symmetry.

5. What are some practical applications of tensor decomposition, Sym representations, and irreps?

Tensor decomposition, Sym representations, and irreps have various practical applications in fields such as data analysis, machine learning, physics, chemistry, and mathematics. They can be used to simplify complex data structures, study symmetry and group theory, and solve various mathematical problems.

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