Symmetry breaking · research workbench

SU(3) branching by a chosen map

Restrict any supported SU(3) highest weight along the regular or principal SU(2) map.

Scope: Regular SU(2) inclusion or principal map with SO(3) image. SU(3) labels p,q from 0–6. Complete SU(2) decomposition; the principal map has kernel Z₂.

Exact resultExact checks passed

\left.V_{(1,1)}\right|_{\iota_{\mathrm{pr}}}=V_{2}\oplus V_{4}

Map image: SO(3)
Map kernel: $\mathbb Z_2$
Parent dimension: $8$
SU(2) singlets: $0$

\operatorname{diag}(z,z^{-1})\longmapsto\operatorname{diag}(z^2,1,z^{-2})

Restriction to SU(2); V_n has dimension n+1
SU(2) highest weight n	Spin j=n/2	Multiplicity	Dimension	SO(3) descent
$2$	$1$	$1$	$3$	Allowed
$4$	$2$	$1$	$5$	Allowed

The principal homomorphism has kernel Z₂ and image SO(3). Every output highest weight is even, so every summand descends to that image. No U(1) charge refinement or basis-dependent intertwiner is included.

✓ dimension conserved✓ character identity✓ nonnegative integer multiplicities✓ kernel compatible

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Method & interpretation

Restrict along the specified torus map

An exact SU(3) weight character is restricted to the chosen SU(2) torus. Differences of adjacent weight multiplicities determine the SU(2) summands. Kernel compatibility is checked.

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Sources, conventions, and version

Computed locally with exact integer or algebraic arithmetic. Independent human expert review is pending. The export records the conventions and implementation version.

Lie Groups and Lie Algebras ISections 25.3–25.4, 28.1–28.4: characters, Casimir eigenvalues, Weyl dimensions and type-A representations.
SageMath: The Weyl Character RingCoroot notation; tensor products; symmetric and exterior powers; Weyl dimension formula; SL versus GL.
SageMath: Gelfand–Tsetlin patternsGelfandTsetlinPattern.weight(): interlacing arrays and differences of row sums.
Physics 251: Solution Set 4Printed pages 14–15: regular SU(2) and SO(3) subgroups of SU(3).

Convention: su3-principal-su2-v1
Algorithm: lie-representations-1.0.0

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