The Littlewood–Richardson method for SU(N)

The SU(N) tensor workbench decomposes a tensor product into irreducible representations with exact multiplicities. Its mathematical category is finite-dimensional continuous complex representations of the compact simply connected group $SU(N)$, in characteristic zero. The calculation uses polynomial characters of $GL(N,\mathbb C)$ and then restricts to determinant one. The global group, representation field, and restriction step are part of the convention.

The convention identifier is su-n-young-lr-v1; the algorithm identifier is su-n-tensors-1.0.0. The implementation is original browser-local code. SageMath and the papers below provide mathematical references and independent fixtures; this workbench does not call an external algebra service.

Dynkin labels and canonical partitions

Order the $A_{N-1}$ nodes along the chain, with node $1$ giving the defining representation. An input is a vector of nonnegative Dynkin labels

$$a=(a_1,\ldots,a_{N-1}),\qquad \lambda_i=\sum_{j=i}^{N-1}a_j,\qquad \lambda_N=0.$$

The weakly decreasing row lengths $\lambda=(\lambda_1,\ldots,\lambda_N)$ specify a Young diagram in English convention: longest row at the top. Adjacent differences recover the labels, $a_i=\lambda_i-\lambda_{i+1}$. This selects one polynomial $GL(N)$ lift of an $SU(N)$ irrep. Adding the same integer to every row tensors the lift with a determinant power and leaves its restriction to $SU(N)$ unchanged. See SageMath, “Coroot notation” and “SL versus GL”.

For example, the $SU(5)$ labels $(0,1,0,0)$ give $(1,1,0,0,0)$, the two-index alternating representation of dimension $10$. Its conjugate has labels $(0,0,1,0)$ and canonical partition $(1,1,1,0,0)$. Equal dimensions do not identify these two representations.

What the coefficient counts

For partitions $\lambda,\mu$, Schur characters multiply as

$$s_\lambda(x)s_\mu(x) =\sum_\nu c^{\nu}_{\lambda\mu}\,s_\nu(x).$$

The Littlewood–Richardson coefficient is zero unless $\lambda\subseteq\nu$ and $|\nu|=|\lambda|+|\mu|$. Otherwise it counts fillings of the skew diagram $\nu/\lambda$ with exactly $\mu_i$ entries equal to $i$, subject to three conditions:

1. Entries weakly increase from left to right along each row.
2. Entries strictly increase down each column.
3. Read each row from right to left, starting with the top row. In every initial segment, the number of $i$ entries is at least the number of $i+1$ entries, for every $i$.

These conditions fix the reading convention completely. The third condition is the lattice-word condition. A precise statement appears in Stanley, Symmetries of Plane Partitions, §3, Theorem 3.2. The official SageMath lrcalc interface supplies independently reproducible coefficient and product examples.

The implementation traverses the skew cells in that reading order. A proposed entry is rejected immediately if it violates a row, column, content, or prefix condition. Every accepted complete filling contributes one. The symmetry $c^\nu_{\lambda\mu}=c^\nu_{\mu\lambda}$ allows the smaller content to be filled first. This is an enumeration of the defining combinatorial objects, with no numerical fitting of dimensions.

Returning from GL(N) to SU(N)

Only partitions with at most $N$ rows contribute in $N$ variables. A longer diagram gives the zero Schur polynomial, because a column of length greater than $N$ cannot be filled strictly with $N$ symbols. For each surviving partition, pad to $N$ rows and set

$$k=\nu_N,\qquad \widehat\nu_i=\nu_i-k,\qquad b_i=\nu_i-\nu_{i+1}.$$

Then

$$S_\nu\mathbb C^N \cong (\det)^k\otimes S_{\widehat\nu}\mathbb C^N, \qquad V_a\otimes V_{a'} \cong\bigoplus_\nu c^\nu_{\lambda\mu}\,V_b.$$

A full column of height $N$ therefore disappears from the $SU(N)$ label. The output preserves both the original $GL(N)$ partition and the number $k$ of removed columns. Distinct output partitions of the same total size cannot collapse to the same Dynkin labels: their difference would have to be a nonzero determinant shift, which changes the size by a nonzero multiple of $N$. Multiplicity is retained in $c^\nu_{\lambda\mu}$.

For a concrete diagnostic, the fundamental times its dual contains the partition $(1^N)$. It is the determinant representation of $GL(N)$ and the singlet of $SU(N)$. Discarding every diagram with $N$ rows would incorrectly remove this singlet. The determinant distinction is developed in the SageMath Weyl-character tutorial.

Exact checks and their scope

Every result must pass all of the following identities before it is returned.

Check	Exact requirement
Dimension	$\dim V_a\,\dim V_{a'}=\sum_b m_b\dim V_b$
Center charge	$t(b)=t(a)+t(a')\pmod N$ for every summand
Dual singlet	$m_{\mathbf1}=1$ precisely when $a'_i=a_{N-i}$; otherwise it is zero
Character specialization	$s_\lambda(1,2,\ldots,N)s_\mu(1,2,\ldots,N)=\sum_\nu c^\nu_{\lambda\mu}s_\nu(1,2,\ldots,N)$

The dimension is evaluated independently using the type-A Weyl product

$$\dim V_a=\prod_{1\leq i<j\leq N} \frac{\lambda_i-\lambda_j+j-i}{j-i}.$$

Integer numerators and denominators are reduced exactly. Dimensions are exported as decimal strings, so JSON consumers cannot silently round a large value. Tableau counts use arbitrary-precision integers internally and are converted to numeric multiplicities only after an exact safe-integer check.

The central generator $z=e^{2\pi i/N}\mathbf1_N$ acts by $e^{2\pi i t(a)/N}$, with

$$t(a)=|\lambda|=\sum_{i=1}^{N-1}i\,a_i\pmod N.$$

This follows directly because each defining-representation tensor index contributes one power of $z$; removing a determinant column subtracts $N$. The lattice interpretation is given by Conrad, Centers of Simply Connected Semisimple Groups, pp. 1–2.

The extra character check uses the alternant formula

$$s_\nu(x_1,\ldots,x_N) =\frac{\det(x_i^{\nu_j+N-j})_{i,j=1}^N} {\det(x_i^{N-j})_{i,j=1}^N},$$

with integer determinants and exact division. It is independent of the skew-tableau traversal. The variables are distinct, so the denominator is nonzero. Because their product is not one, the check must retain the $GL(N)$ partitions: $s_\nu(x)=(x_1\cdots x_N)^k s_{\widehat\nu}(x)$. Schur polynomial operations and specializations are documented in SageMath’s Schur reference.

One successful specialization is a consistency check, not a separate proof of the full character identity. The completeness of the decomposition rests on the Littlewood–Richardson rule and exhaustive traversal. The verification suite separately compares small Schur values with direct semistandard-tableau sums and compares 987 $SU(3)$ products with the site’s independent weight-character algorithm.

Bounds, failure behavior, and reproducibility

The public tool accepts $2\leq N\leq6$, exactly $N-1$ labels per factor, and $0\leq a_i\leq12$. It also requires

$$|\lambda|+|\mu| =\sum_{i=1}^{N-1}i(a_i+a'_i)\leq16.$$

Both bounds apply. Output labels may exceed the input label bound. The box bound refers to these canonical polynomial lifts; it is neither a dimension cutoff nor a bound invariant under conjugating both factors. It controls the size of the tableau search.

A defensive limit of 2,000,000 visited tableau states applies to a product calculation. The declared domain lies below this limit even without cached coefficients: symmetry leaves at most eight cells to fill for each coefficient; there are 1,116 possible lattice-word prefixes through length eight and at most 231 candidate output partitions. Thus 257,796 states already bounds the traversal before imposing the additional content, row, and column restrictions. The verification suite checks this bound.

Exceeding the defensive limit raises an explicit resource error and returns no partial decomposition. Invalid ranks, fractional or negative labels, malformed vectors, and out-of-domain diagrams also fail explicitly. The checked domain is not enlarged silently.

The verification suite covers all 11,666 admissible input pairs at $N=6$, the full supported $SU(2)$ angular-momentum rule, horizontal and vertical Pieri rules, associativity examples, conjugation, determinant shifts, and the published $SU(4)$–$SU(6)$ examples in Feger–Kephart, LieART, Tables A.37–A.39. To reproduce a nontrivial multiplicity in SageMath:

import sage.libs.lrcalc.lrcalc as lrcalc
lrcalc.lrcoef([3, 2, 1], [2, 1], [2, 1])  # 2
lrcalc.mult([2, 1], [2, 1], 3)  # GL(3) partitions, at most 3 rows

Sage’s coefficient function places the output partition first. This site’s helper uses littlewoodRichardson(lambda, mu, nu) instead. The full $SU(3)$ restriction of the displayed product is $\mathbf8\otimes\mathbf8=\mathbf1\oplus2\mathbf8\oplus\mathbf{10}\oplus\overline{\mathbf{10}}\oplus\mathbf{27}$.

What a decomposition does not specify

The answer determines abstract irreducible summands and their multiplicity spaces. It does not choose normalized Clebsch–Gordan maps, a basis inside a repeated summand, or the permutation symmetry of identical factors. A self-dual label is further distinguished as real or quaternionic using the Frobenius–Schur indicator; the relevant general result is Adams, The Real Chevalley Involution, Lemma 5.3.

Descent to a quotient of $SU(N)$, Lorentz contractions, flavor symmetry, and Bose or Fermi statistics remain additional parts of a physics calculation. The worked guide explains how to keep these questions separate while using the same output. Source verification and automated checks are recorded; independent expert review remains pending.