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SU(3) tensor products

Resolve a product into irreducible highest weights, retaining conjugates and repeated multiplets.

Scope: Finite-dimensional continuous complex SU(3) representations. Each Dynkin label of either input is 0–6. Complete decomposition within that input domain; output labels may be larger.

Exact resultExact checks passed

V_{(1,1)}\otimes V_{(1,1)}=V_{(0,0)}\oplus 2\,V_{(1,1)}\oplus V_{(3,0)}\oplus V_{(0,3)}\oplus V_{(2,2)}

Input dimensions: $8\times 8$
Product dimension: $64$
Singlet multiplicity: $1$
Total triality: $0\pmod3$

Complete SU(3) tensor-product decomposition
Highest weight	Multiplicity	Dimension	Conjugate	Reality
$(0,0)$	$1$	$1$	$(0,0)$	Real
$(1,1)$	$2$	$8$	$(1,1)$	Real
$(3,0)$	$1$	$10$	$(0,3)$	Complex
$(0,3)$	$1$	$10$	$(3,0)$	Complex
$(2,2)$	$1$	$27$	$(2,2)$	Real

Repeated irreps have a separate multiplicity. The highest weights distinguish conjugates and inequivalent irreps of equal dimension. This is the full tensor product; no exchange-symmetry projection or choice of Clebsch–Gordan basis is made.

✓ dimension conserved✓ nonnegative integer multiplicities✓ character identity✓ center charge conserved

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Method & interpretation

Multiply full weight characters

Integer Gelfand–Tsetlin patterns construct both characters. Highest-weight subtraction decomposes their product; the full character, dimension, and triality must agree.

Read the worked guide · Mathematical method

Download script for an independent environment; not executed by this website. Requires SageMath; compares complete Weyl characters, dimensions, and singlet multiplicity with the saved result.

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.
Centers of Simply Connected Semisimple GroupsPages 1–4: the center is dual to P/Q, with classical and exceptional types tabulated.

Convention: su3-dynkin-pq-v1
Algorithm: lie-representations-1.0.0

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