SO(N) adjoint rep. under SO(3) subgroup

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SUMMARY

The discussion focuses on the transformation of the SO(N) adjoint representation under the SO(3) subgroup. It highlights the decomposition of the fundamental representation, specifically noting that SO(N) fundamental N results in the expression N ⊗ N = 1 ⊕ A ⊕ S. The participant expresses confusion regarding the S component and seeks clarification on utilizing indices for vector spaces related to tensor products to better visualize the separation of symmetric and antisymmetric parts.

PREREQUISITES
  • Understanding of Lie groups, specifically SO(N) and SO(3).
  • Familiarity with representation theory and tensor products.
  • Knowledge of symmetric and antisymmetric representations in group theory.
  • Basic proficiency in mathematical notation and indices related to vector spaces.
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  • Research the properties of SO(N) representations and their decompositions.
  • Study the role of indices in tensor products and their applications in representation theory.
  • Explore the relationship between symmetric and antisymmetric representations in detail.
  • Examine examples of SO(3) subgroup representations within larger SO(N) groups.
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The discussion is beneficial for theoretical physicists, mathematicians specializing in group theory, and students studying representation theory in the context of Lie groups.

mkgsec
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Hi.

I'm having trouble figuring out how SO(N) adjoint rep. transforms

under a SO(3) subgroup.

Unlike SU(N), SO(N) fundamental N gives

\begin{equation} N \otimes N = 1 \oplus A \oplus S \end{equation}

So the \begin{equation} S \end{equation} part really bothers.

Can you give a help?
 
Last edited:
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What about using indexes for the vector spaces associated to the tensor products? Sometimes it helps to visualize the situation, specially now that you want to separate the symmetric and antisymmetric parts.
 

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