Model building · research workbench II

SU(N) tensor products

Decompose compact SU(N) representations with exact Littlewood–Richardson coefficients and explicit Young-diagram conventions.

Scope: Finite-dimensional continuous complex SU(N) representations; 2≤N≤6. Each Dynkin label 0–12; at most 16 canonical Young boxes across the two inputs. Complete decomposition within the supported domain.

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\begin{aligned}V_{(0,1,0,0)}\otimes V_{(0,1,0,0)}&=V_{(0,0,0,1)}\oplus V_{(1,0,1,0)}\\&\quad{}\oplus V_{(0,2,0,0)}\end{aligned}

Input dimensions: $10\times 10$
Product dimension: $100$
Singlet multiplicity: $0$
Combined Young boxes: 4

Input 1 ·

(1,1)

Input 2 ·

(1,1)

Complete irreducible decomposition
Dynkin labels	Multiplicity	Complex dimension	Conjugate labels	N-ality
$(0,0,0,1)$	$1$	$5$	$(1,0,0,0)$	$4\pmod{5}$
$(1,0,1,0)$	$1$	$45$	$(0,1,0,1)$	$4\pmod{5}$
$(0,2,0,0)$	$1$	$50$	$(0,0,2,0)$	$4\pmod{5}$

Dynkin labels identify each summand, including inequivalent representations of the same dimension. This is the full tensor product; exchange-symmetry projections and Clebsch–Gordan tensors require additional data.

Young shapes and determinant columns

Determinant-column reduction
Highest weight	GL(N) partition	SU(N) partition
$(0,0,0,1)$	$(1,1,1,1,0)$	$(1,1,1,1,0)$
$(1,0,1,0)$	$(2,1,1,0,0)$	$(2,1,1,0,0)$
$(0,2,0,0)$	$(2,2,0,0,0)$	$(2,2,0,0,0)$

One exact polynomial-character specialization; not a separate full-character proof.

✓ nonnegative integer multiplicities✓ dimension conserved✓ center charge conserved✓ schur specialization✓ singlet duality

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Method & interpretation

Littlewood–Richardson coefficients

Highest weights become canonical Young diagrams. Exact skew-tableau multiplicities give the Schur product; determinant columns are then removed for SU(N). The result is checked by dimension, center charge, and an exact character specialization.

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Sources, conventions, and versions

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SageMath: Littlewood–Richardson Calculator interfacelrcoef, mult, coprod: coefficients, Schur products, and coproducts; the [2,1] square fixture.
Symmetries of Plane PartitionsSection 3, Theorem 3.2: skew-tableau formulation of the Littlewood–Richardson rule.
SageMath: Schur symmetric functionsSchur bases, product and coproduct, and specialization as polynomial characters.
SageMath: The Weyl Character RingCoroot notation, SL versus GL, determinant shifts, and tensor decomposition.
Centers of Simply Connected Semisimple GroupsPages 1–2: the center and the quotient of weight and root lattices; type-A center coordinates.
The Real Chevalley InvolutionLemma 5.3: the Frobenius–Schur indicator of a self-dual finite-dimensional irrep.
LieART — A Mathematica Application for Lie Algebras and Representation TheorySection 4.5.3 and Tables A.37–A.39: Young-tableau tensor methods and SU(4), SU(5), SU(6) fixtures.

Convention: su-n-young-lr-v1
Algorithm: su-n-tensors-1.0.0

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