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As your QFT theory text should tell you, for an action invariant under G, addition of a Higgs potential only invariant under its subgroup H will spontaneously break the generators in G/H. You ought to do due diligence and study and understand and reproduce all examples of elementary classics such as Ling-Fong Li, PhysRev D9 (1974) 1723. There are, of ...

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Since you know about $SU(2)$ characters, this is a doable exercise. Let me remind you that the character of spin-$j$ is $$\chi_j(t) = \sum_{m=-j}^j e^{2imt} = \frac{\sin (2j+1)t}{\sin t}.$$ The crucial property is that the character of a tensor product is the product of characters, i.e. $\chi_{j \otimes j'}(t) = \chi_j(t) \chi_{j'}(t).$ In your case, ...

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Short answers Apply the Young calculus (per ACuriousMind's suggestion in the comments). For finding the multiplicity of the trivial representation in a tensor product of representations of $SU(n)$, note that each irreducible representation $D$ of $SU(n)$ has a unique conjugate irreducible representation $\bar D$ such that the Young calculus allows ...

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There is no such a concept like "correct normalization" in Lie algebra representations. You just have to be consistent. By the way, the most used normalization for the adjoint of SU(2) is $$R(T_a)=\frac{1}{2}\sigma_a,$$ which gives $$[R(T_a),R(T_b)]=if_{abc}R(T_c)=i\epsilon_{abc}R(T_c),$$ and $$Tr(R(T_a)R(T_b))=\frac{\delta_{ab}}{2}.$$ Suppose now we use ...

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Any rotation is represented by a rank 3, 3×3 orthonormal matrix. If you agree to this, then all you have to do is prove that a sequence of three rotations yields such a matrix. The problem is that it does not always yield such a result (see gimbal lock). So what you are forced to do is look that there is at lease one set of angles and rotation conventions ...

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