Applying $P[a,b,c]=[b,a,b+c]$ to the eight weights of $d_3$ gives
$$
\begin{gathered}
[0,0,1],\ [1,0,0],\ [-1,1,0],\ [0,1,-1],\\
[0,-1,1],\ [1,-1,0],\ [-1,0,0],\ [0,0,-1].
\end{gathered}
$$
This is exactly the union of the weight sets of $d_1$ and $d_2$. It expresses the <branching rule> for the standard inclusion $D_3\cong A_3\subset B_3$: the eight-dimensional spinor representation of $\mathfrak{so}_7$ restricts to the two four-dimensional chiral spinor representations of $\mathfrak{so}_6\cong\mathfrak{sl}_4$,
$$
\mathbf8\downarrow_{A_3}=\mathbf4\oplus\overline{\mathbf4}.
$$

Solution

Past exam of the mathematics course of the University of Cambridge / 2025 / iii / Paper 302 / 4 / e / Solution

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A branching rule decomposes a representation into irreducible representations after restriction from a Lie group or Lie algebra to a subgroup or subalgebra.


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 Incoming links: Branching rule