What can you build with QRecoupling?
The package is most useful when a calculation needs many related recoupling coefficients, or needs to move between numerical experiments and exact formulas without rewriting the underlying symbol definitions.
Fusion-based tensor networks
Use fmatrix, rmatrix, and bmatrix as local changes of basis and braiding operations. A practical first study is to measure how allowed channel counts and local tensor sizes grow with the level, then compare contractions with and without fusion-sector sparsity. The tensor-network tutorial provides tested local building blocks. A network optimizer or contraction engine is not included.
Spin networks and state-sum models
Use q6j coefficients and quantum dimensions for tetrahedral and edge weights, with your model's phase and normalization conventions. Compare a small network at several levels with its classical limit, or check a local recoupling move before implementing a full state sum. Summing over triangulations, boundary conditions, and observables is application work.
Exact identities and special levels
Explore when a coefficient vanishes, reduces to a rational number, or admits a short radical expression. Symbolic() and x_form expose generic formulas; Exact(k) specializes to the real algebraic basis of a level. Generic polynomial identities can establish a relation wherever its denominators and branch assumptions permit specialization. The identity tutorial separates such proofs from numerical checks.
Deformations and asymptotics
Sweep k toward the classical limit or vary a real/complex q while keeping labels fixed. Batches reuse target tables; scalar BigFloat calculations provide higher-precision checks for difficult points. Near poles and roots of unity, distinguish sensitivity to q from error in the evaluation. Asymptotic fits and geometric interpretations remain the researcher's task.
Finite series and new symbols
Encode factorial-product sums with FactorialSum and test the same expression classically, at a level, and at generic q. This supports terminating hypergeometric examples with integer factorial parameters and offers a starting point for new recoupling formulas. Arbitrary parameterized q-Pochhammer factors and infinite basic hypergeometric series are outside the current interface.
Choosing a first project
| Question | Starting point | Useful validation |
|---|---|---|
| How do fusion spaces grow with k? | fmatrix_labels, QRecoupling.level_labels | Check channel admissibility |
| Are my tensor phases consistent? | F/R matrices and scalar symbols | Orthogonality and local identities |
| Does a relation hold beyond sampled levels? | x_form, prove_identity | Exact polynomial residual and exceptional levels |
| Is a small numerical coefficient really zero? | Exact level or generic formula | Exact cancellation, not only a tolerance |
| Does a custom series recover a classical identity? | FactorialSum, Exact() | Independent factorial/binomial formula |
Performance depends on the labels, target, precision, and workload. Benchmark the intended application rather than extrapolating a scalar timing to a full tensor contraction.