Branching rule

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

A branching rule gives the irreducible representation decomposition after restricting a group representation to a subgroup, or a Lie algebra representation to a Lie subalgebra. For the standard block embedding $SU(5)\subset SU(6)$, $6\downarrow =5\oplus 1$ and ${\Lambda }^{2}6\downarrow =10\oplus 5$.

A coordinate matrix expressing restriction of weights to the Cartan subalgebra of a Lie subalgebra. With Dynkin labels as column vectors, its rows encode the subalgebra's coroots in terms of the original coroots. It need not be an idempotent projection operator: an equal-rank embedding can give an invertible matrix. Applying it to the full weight system, including weight multiplicities, permits reconstruction of the restricted irreducible representations.

For B3 root system simple roots $e_1-e_2,e_2-e_3,e_3$, choose the D3 root system simple roots in chain order $e_2-e_3,e_1-e_2,e_2+e_3$. The Dynkin labels project by $(a,b,c)\mapsto (b,a,b+c)$. This separates the vector weights $\{0,\pm e_i\}$ into the six-dimensional exterior square of the defining SU(4) representation and a singlet representation. The spinor cube vertices $\frac{1}{2}(\pm e_1\pm e_2\pm e_3)$ split into its defining representation and its dual representation.

A root string of length $\Lambda +1$ gives an irreducible sl2 Lie algebra module $R(\Lambda )$ when restricting the Adjoint representation to the sl2 subalgebra associated with a root. Include also the root's own $R(2)$ and one commuting Cartan direction $R(0)$. For $A_2$ either simple root gives $R(2)\oplus 2R(1)\oplus R(0)$. For $B_2$, the long root gives $R(2)\oplus 2R(1)\oplus 3R(0)$ and the short root gives $3R(2)\oplus R(0)$. For $G_2$, the long root gives $R(2)\oplus 4R(1)\oplus 3R(0)$ and the short root gives $R(2)\oplus 2R(3)\oplus 3R(0)$.