Symmetry breaking · research workbench II

Branching with U(1) charges

Restrict an SU(N) representation along a specified block map, keeping both nonabelian labels and the commuting U(1) charge.

Scope: SU(p) × SU(N−p) × U(1) mapped into SU(N); 3≤N≤6, 1≤p<N. At most 12 canonical Young boxes; Dynkin labels 0–12. Primitive U(1) normalization and the finite map kernel are explicit.

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Exact resultExact checks passed

\begin{aligned}\left.V_{(0,1,0,0)}\right|_{\iota}&=\left(V_{(0,0)},V_{(0)}\right)_{-6}\oplus \left(V_{(1,0)},V_{(1)}\right)_{-1}\\&\quad{}\oplus \left(V_{(0,1)},V_{(0)}\right)_{4}\end{aligned}

\operatorname{SU}(3)

\operatorname{SU}(2)

\operatorname{U}(1)_X

SU(5)

\iota(A,B,z)=\operatorname{diag}\!\left(z^{2}A,z^{-3}B\right)

Parent complex dimension: $10$
Map kernel: $\mathbb Z_{6}$
Map image: $S\!\left(U(3)\times U(2)\right)$
Charge normalization: $Q=6Y=-X$

The decomposition subscripts are the primitive charge X. The physical hypercharge column uses Y = −X/6; the product-group workbench uses the integer charge Q = 6Y = −X.

Irreducible multiplets with their commuting U(1) charge
SU(3) labels	SU(2) labels	Multiplicity	Dimensions	X	Y = −X/6
$(0,0)$	$(0)$	$1$	$1\times 1$	$-6$	$1$
$(1,0)$	$(1)$	$1$	$3\times 2$	$-1$	$\frac{1}{6}$
$(0,1)$	$(0)$	$1$	$3\times 1$	$4$	$\frac{-2}{3}$

Kernel generator and charge checks

\left(e^{-2\pi i/3}\mathbf1_{3},e^{2\pi i/2}\mathbf1_{2},e^{2\pi i/6}\right)

−2 t_3 + 3 t_2 + X = 0 mod 6.

\operatorname{Tr}_R X=0,\qquad\operatorname{Tr}_R X^2=90

These traces check representation theory and normalization; they are not a full anomaly analysis.

Use these multiplets in a global-form calculation

The transfer converts X to Q = 6Y = −X and uses the Standard Model orientation of the diagonal generator. Repeated copies give the same center condition; the branching export retains multiplicities.

Check the quotient

✓ dimension✓ map kernel✓ traceless charge✓ quadratic charge trace✓ nonnegative integral multiplicities

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Method & interpretation

Keep the block determinant charges

The Schur coproduct decomposes the parent character on the two blocks. Determinant powers are retained until the primitive U(1) charge is assigned. Dimension, the map kernel, and the first two charge moments are checked.

Read the worked guide · Mathematical method

Sources, conventions, and versions

Exact computations run in this browser. Independent human expert review is pending. Calculation JSON preserves inputs, sources, conventions, and the mathematical engine version.

SageMath: Littlewood–Richardson Calculator interfacecoprod(part, all=1), lrcoef: the full Schur coproduct and its Littlewood–Richardson coefficients.
SageMath: Maximal Subgroups and Branching RulesBranching rules and embeddings of maximal tori: restriction depends on the specified map.
The Algebra of Grand Unified TheoriesSection 3.1: SU(5), exterior powers of its fundamental, and the Standard Model subgroup. Hypercharge conventions must be translated explicitly.
Line Operators in the Standard ModelIntroduction and Section 3: the Z6 quotient, integer-normalized hypercharge, and the additional choice of line operators.

Convention: su-n-block-u1-v1
Algorithm: su-n-block-branching-1.0.0

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