explicit matrix elements for a representation decomposed into subgroup by branching rules
I'm looking for a way to construct a representation for a simple Lie group such that one particular subgroup is manifest. I learned the branching rules from <PERSON>, <PERSON> and <PERSON>, but I'm still not sure how to derive the explicit representation from the weight system deformed by extended <PERSON> diagram?
To decompose a irreducible representation under a maximal regular subalgebra, we look at the extended <PERSON> diagram, which has one more node from the minus highest root added on the original <PERSON> diagram, and we then eliminate another node to deform the extended <PERSON> diagram into the <PERSON> diagram for the maximal regular subalgebra. The key point is to replace the <PERSON> coefficient corresponding to the eliminated node by the one corresponding to the extra node of the minus highest root. Thus, deforming the weight system of the given irrep with above replacement gives us the branching rules and the Cartan matrices for the subalgebra in the given irrep. 364
For example, under the subgroup $SU(2)\times SU(2)\times SU(2) \Subset SO(7)$, we have the 8 dimensional irrep of SO(7) deformed as $8 \rightarrow (1,2,2) + (2,1,2)$, see P33-34 LieART.
To construct the explicit matrix representation, we define the coordinates for each simple roots in an orthonormal basis (Cartan-Weyl basis) and therefore the coordinates for the fundamental weights as well. Thus, the generators corresponding to Cartan matrices have its matrix entries as the coordinate of every weights. Now, my question is what are the fundamental weights for the deformed <PERSON> diagram? Should I keep the old fundamental weights except the replaced one or the new fundamental weights are resolved from the deformed <PERSON> diagram? Most importantly, what's the new fundamental weight corresponding to the extended root?
I would also need help to reveal how generators of the subalgebra are in the subset of generators of the mother algebra. When I replace the <PERSON> coefficient with the extended root, how could the corresponding Cartan matrix belong to linear combination of Cartan matrices of the mother algebra?
I'm a Physics student with poor knowledge in Lie algebra. Please reinterpret my description with proper language whenever you needed to. 364