Two SU(2) maps into SU(3)

A branching rule belongs to a particular homomorphism. For $SU(3)$, two maps with domain $SU(2)$ already produce different answers:

$$\mathbf3\downarrow_{\mathrm{regular}}=\mathbf2\oplus\mathbf1, \qquad \mathbf3\downarrow_{\mathrm{principal}}=\mathbf3.$$

The regular map is injective. The principal map has kernel $\{\pm I\}$ and image isomorphic to $SO(3)$. The maps, their torus restrictions, and their kernels are retained in every calculation export. These two subgroup constructions are exhibited in Haber, Physics 251 Solution Set 4, pp. 14–15.

Open the branching workbench

The maps and their global meaning

Let $z=e^{i\theta}$ with $\theta$ of period $2\pi$ in the standard $SU(2)$ torus.

Property	Regular	Principal
Homomorphism	$\iota_{\mathrm r}(U)=\operatorname{diag}(U,1)$	$\iota_{\mathrm p}(U)=\operatorname{Sym}^2(U)$
Torus image	$\operatorname{diag}(z,z^{-1},1)$	$\operatorname{diag}(z^2,1,z^{-2})$
Kernel	$\{I\}$	$\{I,-I\}$
Image in $SU(3)$	A subgroup isomorphic to $SU(2)$	A subgroup conjugate to the real $SO(3)$ subgroup
Trace embedding index	$1$	$4$

The symmetric square is written in an orthonormal basis of the spin-one representation. Its determinant is one. The element $-I\in SU(2)$ acts trivially on that square, explaining the principal kernel.

The two maps have injective differentials at the Lie-algebra level. Only one is an embedding of the stated global group $SU(2)$. The principal map factors through

$$SU(2)\longrightarrow SO(3)\hookrightarrow SU(3).$$

The atlas records retain these separate facts: regular map and principal map.

Labeling the restricted multiplets

An $SU(2)$ irrep $V_n$ has highest weight $n=2j$ and complex dimension $n+1$. Its torus character is

$$\chi_n(z)=z^n+z^{n-2}+\cdots+z^{-n}.$$

The center acts by $(-1)^n$. Every principal restriction must therefore contain only even $n$, or equivalently only integer spins. The workbench checks this kernel condition independently of the dimension sum.

Here are several restrictions. Bold numerals label dimensions; repeated summands denote independent copies.

$SU(3)$ irrep $(p,q)$	Regular restriction	Principal restriction
$\mathbf3=(1,0)$	$\mathbf2\oplus\mathbf1$	$\mathbf3$
$\overline{\mathbf3}=(0,1)$	$\mathbf2\oplus\mathbf1$	$\mathbf3$
$\mathbf6=(2,0)$	$\mathbf3\oplus\mathbf2\oplus\mathbf1$	$\mathbf5\oplus\mathbf1$
$\mathbf8=(1,1)$	$\mathbf3\oplus2\mathbf2\oplus\mathbf1$	$\mathbf5\oplus\mathbf3$
$\mathbf{10}=(3,0)$	$\mathbf4\oplus\mathbf3\oplus\mathbf2\oplus\mathbf1$	$\mathbf7\oplus\mathbf3$
$\mathbf{27}=(2,2)$	$\mathbf5\oplus2\mathbf4\oplus3\mathbf3\oplus2\mathbf2\oplus\mathbf1$	$\mathbf9\oplus\mathbf7\oplus2\mathbf5\oplus\mathbf1$

The first row fixes the maps; the remaining rows follow by restriction of the complete characters. Several have short independent derivations.

For the six-dimensional representation, use $\mathbf6=\operatorname{Sym}^2\mathbf3$. The regular restriction gives

$$\operatorname{Sym}^2(\mathbf2\oplus\mathbf1) =\operatorname{Sym}^2\mathbf2\oplus\mathbf2\oplus\mathbf1 =\mathbf3\oplus\mathbf2\oplus\mathbf1.$$

For the principal map, $\operatorname{Sym}^2\mathbf3$ decomposes into a symmetric traceless rank-two tensor and its trace, giving $\mathbf5\oplus\mathbf1$.

For the adjoint, restrict $\mathbf3\otimes\overline{\mathbf3}=\mathbf1\oplus\mathbf8$ and remove one trivial summand. This yields both eight-dimensional rows above using only the $SU(2)$ Clebsch–Gordan rule. Reproduce the principal $\mathbf6$ restriction.

How the complete restriction is calculated

The input irrep $(p,q)$ is represented by the polynomial $U(3)$ highest weight $(p+q,q,0)$ and then restricted to $SU(3)$. Its character can be enumerated with interlacing Gelfand–Tsetlin patterns:

$$p+q\geq u\geq q\geq v\geq0, \qquad u\geq w\geq v.$$

Each pattern contributes the torus weight

$$(w_1,w_2,w_3) =(w,\ u+v-w,\ p+2q-u-v).$$

The multiplicity of a weight counts how many patterns produce it. These are the row-sum weight coordinates described in SageMath, Gelfand–Tsetlin patterns, weight method.

Substitute the chosen torus map. A monomial contributes exponent

$$m_{\mathrm r}=w_1-w_2, \qquad m_{\mathrm p}=2(w_1-w_3).$$

After summing equal exponents, write the restricted character as $\sum_m c_m z^m$. Since every $V_n$ contributes one copy of weights $n,n-2,\ldots,-n$, its multiplicity is simply

$$\boxed{N_n=c_n-c_{n+2},\qquad n\geq0.}$$

The code reconstructs the entire restricted character from these $N_n$. It also checks that all multiplicities are nonnegative integers and that

$$\sum_n N_n(n+1) =\frac{(p+1)(q+1)(p+q+2)}2.$$

For the principal map, the even exponents and even highest weights make the $SO(3)$ image visible in the calculation itself.

What the embedding index means

The index here compares the trace in the defining $\mathbf3$ of $SU(3)$ with the trace in the defining $\mathbf2$ of $SU(2)$:

$$\operatorname{tr}_{\mathbf3}\!\bigl(d\iota(X)d\iota(Y)\bigr) =I_\iota\operatorname{tr}_{\mathbf2}(XY).$$

Choose the Hermitian generator $X=\operatorname{diag}(\tfrac12,-\tfrac12)$. Its square has trace $\tfrac12$. Under the regular map the trace stays $\tfrac12$. Under the principal map the eigenvalues are $1,0,-1$, so the trace of the square is $2$. Hence $I_{\mathrm r}=1$ and $I_{\mathrm p}=4$.

This ratio matters when translating a normalized invariant form through a map. It does not change the integer branching multiplicities. Gauge-coupling or charge conventions should be stated separately when the restriction is used in a physical model.

Scope and use in physics

The executable domain is every $SU(3)$ highest weight with $0\leq p,q\leq6$, along the two maps above. Results are calculated locally from exact characters. The site does not call SageMath to produce them; the SageMath branching documentation is an independent reference for the character-restriction approach.

The current output resolves $SU(2)$ multiplets. It does not attach the additional $U(1)$ charges available for a regular $SU(2)\times U(1)$ description. Adding those charges requires a specified group map, its discrete kernel, and a charge normalization.

A restriction can describe how a multiplet transforms under a proposed unbroken symmetry. Establishing an actual symmetry-breaking pattern additionally requires a vacuum and its stabilizer. A singlet in the restriction is useful evidence about invariant directions, but does not by itself determine the full stabilizer, the potential, or the spectrum of physical masses.

For a research calculation, retain the map identifier, the parent labels, the $SU(2)$ labels $n=2j$, and the exported convention profile. Reproduce and cite a calculation.