Block branching with U(1) charges

A branching rule needs a group homomorphism. Naming the two nonabelian factors is insufficient when a commuting $U(1)$ survives: dropping that factor erases charges, can combine distinct multiplets, and obscures which representations descend to the actual subgroup.

Open the charged branching workbench. It computes the complete restriction along the block map defined below. The released domain is $3\leq N\leq6$, $1\leq p<N$, and highest weights with at most twelve boxes in their canonical Young diagram. Representations are finite-dimensional continuous complex representations of compact groups. The cutoff limits the implementation; the formulas apply more generally.

The map and its global image

Write

$$N=p+q,\qquad g=\gcd(p,q),\qquad \alpha=\frac qg,\qquad \beta=-\frac pg.$$

We use the homomorphism

$$\begin{aligned} \iota:\;SU(p)\times SU(q)\times U(1)&\longrightarrow SU(N),\\ (A,B,z)&\longmapsto \operatorname{diag}(z^\alpha A,z^\beta B). \end{aligned}$$

The determinant is one because $p\alpha+q\beta=0$. Here $z=e^{i\theta}$ has period $2\pi$. Since $\alpha$ and $\beta$ are coprime integers, the corresponding diagonal generator

$$X=\operatorname{diag}(\alpha\mathbf1_p,\beta\mathbf1_q)$$

is primitive: its one-parameter subgroup closes for the first time at $\theta=2\pi$. This fixes an integer charge convention before any representation is decomposed. Rescaling $X$ changes both the numerical charges and the period assigned to the parameter.

The image is

$$H=S(U(p)\times U(q)) =\{\operatorname{diag}(U,V):\det U\det V=1\}.$$

To see surjectivity onto $H$, choose an $l$th root $z$ of $\det U$, where $l=pq/g$. Then $A=z^{-\alpha}U$ and $B=z^{-\beta}V$ both have determinant one. This construction also identifies the ambiguity in the preimage.

An element lies in the kernel exactly when

$$A=z^{-\alpha}\mathbf1_p,\qquad B=z^{-\beta}\mathbf1_q,\qquad z^l=1.$$

Consequently,

$$\begin{aligned} \ker\iota&=\langle\gamma\rangle\cong\mathbb Z_l,\\ \gamma&=\left( e^{-2\pi i/p}\mathbf1_p, e^{2\pi i/q}\mathbf1_q, e^{2\pi i/l}\right),\\ H&\cong \frac{SU(p)\times SU(q)\times U(1)}{\mathbb Z_l}. \end{aligned}$$

The generator has order $l$ because its $U(1)$ component does. When a block has size one, $SU(1)$ means the trivial group: its Dynkin-label list is empty and its representation dimension is one. The formula still applies. For example, $S(U(N-1)\times U(1))$ is isomorphic to $U(N-1)$, not to the direct product $SU(N-1)\times U(1)$.

This explicit map is the mathematical object used by the calculator. The importance of specifying the torus map in a branching calculation is discussed in SageMath’s branching tutorial, in its opening “Branching rules” section.

From Dynkin labels to the Schur coproduct

For an $SU(N)$ highest weight $(a_1,\ldots,a_{N-1})$, choose the canonical partition

$$\lambda_i=\sum_{j=i}^{N-1}a_j\quad(i<N),\qquad \lambda_N=0.$$

Its size is $|\lambda|=\sum_{j=1}^{N-1}j a_j$. The polynomial $GL(N,\mathbb C)$ representation with this partition restricts to the desired $SU(N)$ irrep. Adding the same integer to all $N$ rows tensors by a determinant power and gives the same $SU(N)$ representation.

Split the variables of its Schur character into two alphabets. The coproduct identity is

$$s_\lambda(x_1,\ldots,x_p,y_1,\ldots,y_q) =\sum_{\mu,\nu}c^\lambda_{\mu\nu} s_\mu(x_1,\ldots,x_p)s_\nu(y_1,\ldots,y_q).$$

Here $c^\lambda_{\mu\nu}$ is a Littlewood–Richardson coefficient, $|\mu|+|\nu|=|\lambda|$, and the terms vanish unless $\mu$ and $\nu$ have at most $p$ and $q$ rows. The same coefficients that govern products of Schur functions govern this restriction. The SageMath lrcalc documentation describes both operations and exposes the full coproduct through coprod(part, all=1).

Homogeneity now supplies the abelian charge without a separate guess:

$$\begin{aligned} s_\mu(z^\alpha x)&=z^{\alpha|\mu|}s_\mu(x),\\ s_\nu(z^\beta y)&=z^{\beta|\nu|}s_\nu(y),\\ X_{\mu\nu}&=\frac{q|\mu|-p|\nu|}{g}. \end{aligned}$$

The nonabelian highest weights are the adjacent differences

$$b_i=\mu_i-\mu_{i+1},\qquad c_j=\nu_j-\nu_{j+1},$$

after padding the partitions to lengths $p$ and $q$. Each output therefore has an unambiguous identity

$$(b_1,\ldots,b_{p-1});\quad (c_1,\ldots,c_{q-1});\quad X_{\mu\nu},$$

with multiplicity $c^\lambda_{\mu\nu}$. A dimension pair alone does not identify that multiplet.

Keep determinant powers until the charge is known

Removing a full column of height $p$ from $\mu$ leaves its $SU(p)$ representation unchanged but changes $|\mu|$. It therefore changes the charge if the other data are held fixed. Such a column is a charged determinant representation of the block’s $U(p)$, even though it is trivial under $SU(p)$.

In contrast, adding a full column of height $N$ to the parent shifts the contributing block partitions by full columns of heights $p$ and $q$ together. The change in charge is

$$\Delta X=\frac{qp-pq}{g}=0.$$

Thus the result depends only on the parent $SU(N)$ highest weight. The workbench records the intermediate block partitions in its JSON export so this distinction can be inspected.

The quotient condition on every output

Let $t_p$ and $t_q$ be the center charges of the two nonabelian irreps, computed as their Young-box counts modulo $p$ and $q$. The kernel generator acts on a product multiplet by

$$\exp\!\left[2\pi i\left( -\frac{t_p}{p}+\frac{t_q}{q}+\frac Xl \right)\right].$$

It descends to $H$ precisely when

$$-\frac qg\,t_p+\frac pg\,t_q+X=0\pmod l.$$

Every output of a genuine restriction must satisfy this condition. Substituting the charge formula makes the cancellation explicit. A proposed product-group multiplet can fail it even if its two nonabelian representations and its integer $U(1)$ charge are individually valid.

For $p=3,q=2$, the condition is $-2t_3+3t_2+X=0\pmod6$. In the Standard Model convention used on this site, $Q_Y=6Y=-X$, so it becomes

$$2t_3+3t_2+Q_Y=0\pmod6.$$

The SU(5) worked page derives that conversion. The product-group workbench can test the corresponding embedded quotient directly. A compatible matter action alone need not select the full quotient; this distinction is developed in Tong’s discussion of Standard Model line operators, Introduction and Section 3.

The transfer uses one matter row per distinct charged multiplet. Repeated copies have identical center phases, so they impose the same descent condition. Preserve the original branching export when multiplicities are needed for field content or another physical calculation.

Three checks with different failure modes

First, a complete decomposition must conserve dimension:

$$\dim V_\lambda =\sum_{\mu,\nu}c^\lambda_{\mu\nu} \dim V_b^{SU(p)}\dim V_c^{SU(q)}.$$

Second, $X$ is a traceless element of the parent simple Lie algebra, so its trace vanishes in any finite-dimensional representation. The dimensions $d_s$, multiplicities $m_s$, and charges $X_s$ must obey

$$\sum_s m_s d_sX_s=0.$$

Third, use $T(\mathbf N)=\tfrac12$ and the same quadratic-index convention as the Weyl reference. Then

$$\begin{aligned} \operatorname{tr}_{\mathbf N}X^2&=p\alpha^2+q\beta^2=\frac{pqN}{g^2},\\ \sum_s m_s d_sX_s^2&=2T(V_\lambda)\frac{pqN}{g^2}. \end{aligned}$$

The runtime checks dimension, kernel compatibility, and both charge moments exactly. Dimension alone would not detect a reversed charge, and a first moment alone would not detect a uniform rescaling. The development tests additionally compare exact character evaluations constructed independently through the Jacobi–Trudi determinant, test exterior and symmetric powers, and compare charge-forgotten $SU(3)$ results with the separate restriction engine. These tests support the declared implementation; named human scientific review remains pending.

Reproduce a small example

For the $SU(5)$ ten, use the parent partition $(1,1)$ and the $3+2$ split. A SageMath installation can expose the polynomial block terms with:

from sage.libs.lrcalc.lrcalc import coprod

for (mu, nu), multiplicity in coprod([1, 1], all=1).items():
    if len(mu) <= 3 and len(nu) <= 2:
        X = 2 * sum(mu) - 3 * sum(nu)
        print(mu, nu, multiplicity, X)

The all=1 argument matters: the default coproduct display suppresses one member of some exchanged pairs. Convert the printed partitions to adjacent Dynkin differences only after computing $X$. The website runs its own exact JavaScript implementation; the external SageMath recipe is not a live backend call.

The export names su-n-block-u1-v1, the mathematical engine version, the parent weight, the block split, the kernel, and the complete output. It supplies abstract multiplets and multiplicities. A choice of basis, explicit intertwiner, vacuum, mass spectrum, or normalized interaction tensor requires additional information.