Recoupling symbols and conventions

Symbols and label order

These functions use classical values by default. Level(k), Exact(), Exact(k), At(q), and Symbolic() select the other supported regimes. The graph helpers are currently qualified names rather than exports.

using QRecoupling
k = 5
(qdim(1; k=k), q3j(1,1,1,1,-1,0; k=k),
 fsymbol(1,1,1,1,1,1; k=k), gsymbol(1,1,1,1,1,1; k=k))

(2.246979603717467, 0.5974076228942927 + 0.0im, 0.19806226419516182, 1.0000000000000004)

The F convention is

\[[F^{abc}_d]_{ef}=(-1)^{a+b+c+d} \sqrt{[2e+1][2f+1]}\begin{Bmatrix}a&b&e\\c&d&f\end{Bmatrix}_q.\]

Keep the returned channel labels when using fmatrix: they specify the matrix ordering. At a unitary level the matrix is real orthogonal. At generic complex q the corresponding identity uses transpose, not conjugate transpose. Do not assume that a complex recoupling matrix is unitary.

Exact values and phases

v = q6j(Exact(10), 1,1,1,1,1,1)
v

Exact value at level k = 10   (x = 2cos(π/12), degree 4 over ℚ)
  = (2x² − 7)/3

Exact level recoupling coefficients use ExactX, in the real algebraic variable x = 2cos(π/(k+2)), with square-root factors as needed. They do not default to the old cyclotomic-field wrapper. See Exact forms. q3j and qcg have no Exact(k) or Symbolic() form yet: their level values are complex, in ℚ(ζ) up to a square root. Use Level(k) for them.

Braiding and twists carry q phases. rmatrix(Symbolic(), ...) and exact level phase calls retain a QPhase; these are not ordinary real x-polynomials.

phase = rmatrix(Symbolic(), 1//2, 1//2, 1)
phase

Exact phase over x = q + q⁻¹
  = u^(1//2)
  u² − x·u + 1 = 0, with u = q; fractional powers retain the q-phase branch

The display makes the algebraic extension explicit: u² − xu + 1 = 0, with u = q. It retains the branch needed to distinguish q from its inverse. At a numerical target:

(rmatrix(Level(5), 1//2, 1//2, 1), twist(Level(5), 1//2))

(0.9749279121818236 + 0.2225209339563144im, 0.7818314824680298 + 0.6234898018587335im)

Level phases have complex output, including when the target specifies a real precision type such as Level(5; T=BigFloat). Both phase functions also accept a range of levels. The conventions obey rmatrix(a,b,c)^2 = twist(c)/(twist(a)*twist(b)) on admissible channels, with the same evaluation target on each side. tmatrix includes the framing phase by default; use anomaly=false for the diagonal matrix of twists alone.

Generic complex square roots follow the package's balanced branch convention. Replacing products of roots by a principal root of their product can change a sign. Use the supplied symbol evaluators to preserve that convention.

Negative real q uses the same branch as complex(q) throughout the recoupling symbols, coupling coefficients, phases and matrices; the result type is complex even when the value is real. At a point on the negative real axis, including either signed imaginary zero, arg(q)=π and the square root of a negative balanced factor is the positive imaginary root. Passing a nonzero imaginary part evaluates that complex parameter without snapping it to the axis.

Admissibility and model boundaries

For recoupling symbols, level-inadmissible labels return zero by convention. A custom factorial series is a different object: a denominator can vanish at that level and produce a pole. QRecoupling.level_admissible(QRecoupling.SixJ(), k, labels...) checks representation constraints without evaluating a coefficient.

A local 6j coefficient is an ingredient of a state sum, not an entire Turaev–Viro or Ponzano–Regge invariant. Edge weights, summation ranges, triangulation data, normalization, and any boundary observables must be supplied by the application.