I am currently studying representation theory. In my courses notes, I face trouble understanding one step of a concrete example: three masses connected by springs. This is the same approach as in Georgi p27 http://mural.uv.es/rusanra/Lie%20Algebras%20in%20Particle%20Physics%202%C2%AA%20ed%20-%20From%20Isospin%20to%20Unified%20Theories%20(Georgi,%201999).pdf

My problem happen when we have to find the normal modes associated to $D_2$. This point is not fully developped in Georgi. However, in my courses notes, I find this explanation:

"Since $D_6 = D^{def} \otimes D_2$ and that $D^{def} = D_1 \oplus D_2$, we have $D_6 = [D_1 \oplus D_2] \otimes D_2 = (D_2 \otimes D_2) \oplus (D_2 \otimes D_2) = D_2 \oplus ( D_2 \otimes D_2)$.

We can then see that the direction in $\mathbb{R}^3$ invariant under $D^{def}$ tensorized with $\mathbb{R}^2$ transforms under $D_2$."

From this fact, he deduce that $(1,0,1,0,1,0)$ for instance must be a normal mode. I understand all of that except the last sentence: Why is this direction transforming like $D_2$ ? ( I guess transforms under ... mean is an invariant subspace under ... ).



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