racah-py

AI-generated, for agentic coding. This document was written by an AI agent as reference material for AI agents (and humans) working on this repository. It may contain errors — check it against the code and tests rather than trusting it blindly.

Python bindings (PyO3 + maturin) for the racah crate. racah is the coefficient authority behind TeNeT-py's symmetry providers; this package is how that consumer (and any other Python code) reaches it.

Two surfaces:

• SU(N), taking Irrep labels and running the generated Gelfand–Tsetlin pipeline — irreps from Dynkin labels, fusion with outer multiplicities, dense m-basis Clebsch–Gordan tensors, F/R symbols with their multiplicity axes, and the verification gates.
• exact SU(2), taking doubled integer spins and returning scalars from closed-form big-rational arithmetic — wigner_3j, wigner_6j, su2_clebsch_gordan, su2_f_symbol, su2_r_symbol, su2_frobenius_schur.

SU(2) is reachable through either (Irrep([2j]) is a rank-1 label), and they are separate authorities with separate fingerprints. They agree on F and R and differ on the Clebsch–Gordan coefficients by one sign per fusion channel — see "Versioning and the gauge fingerprint" below.

Import name is racah; the distribution is racah-py. Wheels are always built with the crate's cgc-gen feature on, so the generated coefficients are available out of the box. CGC/F/R arrays come back as NumPy float64 arrays.

Installation

From PyPI (Python ≥ 3.12; prebuilt abi3 wheels for linux-x86_64 and macos-arm64):

pip install racah-py

Or from a checkout (needs a Rust stable toolchain):

git clone https://github.com/Ryo-wtnb11/racah
cd racah/racah-py
python -m venv .venv && . .venv/bin/activate
pip install maturin
maturin develop --release          # builds the extension into the active venv

or build a wheel and install it anywhere:

maturin build --release --manifest-path racah-py/Cargo.toml --out dist
pip install --no-index --find-links dist racah-py

CI (wheels.yml) builds abi3-py312 wheels for linux-x86_64 and macos-arm64; py-v* tags/releases publish them to PyPI. One wheel per platform covers CPython ≥ 3.12.

Quick start

Verified against the built extension; every call below is the actual API (see racah.pyi for the full typed surface).

import racah

three = racah.Irrep([1, 0])            # SU(3) fundamental, from its Dynkin label
assert three.dim() == 3                # exact integer, arbitrary precision
eight = racah.Irrep([1, 1])            # the adjoint

# Fusion: 3 x 3bar = 1 + 8, and the adjoint appears twice in 8 x 8.
assert racah.fusion(three, three.dual()) == [
    (racah.Irrep([0, 0]), 1),
    (eight, 1),
]
assert racah.fusion_multiplicity(eight, eight, eight) == 2  # N^8_88

# m-basis Clebsch-Gordan tensor: [dim(s1), dim(s2), dim(s3), N^{s3}_{s1 s2}].
cgc = racah.clebsch_gordan(three, three.dual(), eight)
assert cgc.shape == (3, 3, 8, 1)

# F-symbol block over the four multiplicity indices [mu, nu, kappa, lambda].
f = racah.f_symbol(eight, eight, eight, eight, eight, eight)
assert f.shape == (2, 2, 2, 2)

# R-symbol multiplicity matrix, N^c_ab x N^c_ba.
r = racah.r_symbol(three, three.dual(), eight)
assert r.shape == (1, 1)

# The verification gates raise RuntimeError on violation.
racah.check_pentagon(three, three.dual(), three, three.dual())

# The gauge fingerprint a consumer pins next to persisted coefficients.
print(racah.sun_authority_fingerprint())

The exact SU(2) surface takes doubled labels (dj = 2j, dm = 2m), so every label stays an exact integer:

import racah

assert abs(racah.wigner_6j(2, 2, 2, 2, 2, 2) - 1 / 6) < 1e-14
assert racah.su2_r_symbol(1, 1, 0) == -1.0        # (-1)^(1/2 + 1/2 - 0)
assert racah.su2_r_symbol(1, 1, 4) == 0.0         # inadmissible is exact zero,
                                                  # not an error
print(racah.su2_authority_fingerprint())

Reaching the same R-symbol through r_symbol(Irrep([1]), Irrep([1]), Irrep([0])) builds a Clebsch–Gordan tensor first; su2_r_symbol is a sign.

The rest of the surface: Irrep.from_weight / Irrep.trivial constructors, the dynkin / weight / rank properties, and check_f_unitarity and check_hexagon alongside check_pentagon.

Ill-posed input (bad label, mixed rank, empty fusion vertex) raises ValueError; a tripped numerical gate (orthonormality, F-unitarity, pentagon, hexagon, factorization failure) raises RuntimeError.

Versioning and the gauge fingerprint

F/R/CGC values depend on the CGC gauge; racah publishes them in one frozen canonical gauge. racah.sun_authority_fingerprint() returns the opaque authority string identifying that convention, generation pipeline, and tolerance policy — TeNeT-py pins it next to persisted coefficients and refuses a mismatch on load. Compare it by equality only; a breaking coefficient change bumps the fingerprint epoch and is recorded in the CHANGELOG. Same fingerprint means same convention and value agreement within the oracle tolerance, not cross-process bit-identity (docs.rs: sun_authority_fingerprint).

racah.su2_authority_fingerprint() is its twin for the exact SU(2) surface, and the two are different strings on purpose. Numerical agreement is not authority identity: F-symbols and R-symbols do agree between the tiers to round-off, but their Clebsch–Gordan coefficients differ by exactly one sign per fusion channel, uniform in the magnetic indices and equal to su2_r_symbol itself. F and R are gauge-invariant combinations in which that phase cancels; CGC are gauge data. Mixing the two tiers' CGC without the factor gives wrong signs and no error, so record the fingerprint of whichever surface produced what you persisted.

Documentation

The semantics are the crate's; the bindings add nothing:

• User Guide — task-oriented chapters; shapes and axis conventions match the arrays returned here.
• docs/theory.pdf — the mathematics behind the objects.
• docs/gauge.md — the frozen SU(N) gauge specification the fingerprint names.
• docs.rs/racah — exact per-item semantics and errors.
• docs/README.md — the full documentation index.

Development

uv venv && uv pip install maturin pytest numpy
cd racah-py && maturin develop --release
pytest tests