[lkdvos]

For GenericFusion sectors, the second (underbraiding) equation in hexagon_equation used conj(R) for the inverse braiding. The inverse of a unitary R-symbol is its adjoint, so the first two indices of Race, Rbcf and Rgcd also need to be swapped. This is the same fix that #92 made in the Artin braid test, and it matches what TensorKit uses for inv = true braids (Rsymbol(b, a, c)' in braidingtensor.jl and fusiontrees/braiding_manipulations.jl).

The two versions differ only when an R-symbol is not symmetric. No sector in this package has such an R-symbol: the only one with multiplicities, the 2×2 R-symbol of A4Irrep, is symmetric. So the existing tests pass either way. SU(3) irreps from SUNRepresentations.jl do have non-symmetric R-symbols. With TensorKitSectors' sector testsuite (QuantumKitHub/SUNRepresentations.jl#44), for example, hexagon_equation(SU3Irrep((4, 1, 0)), SU3Irrep((3, 0, 0)), SU3Irrep((4, 2, 0))) failed with a mismatch of 4e-2 before this PR. For the five SU(3) triples that failed there:

• before this PR: the first (overbraiding) equation holds to 5e-15, and the second fails by 4–7e-2;
• with this PR: both equations hold to 5e-15.

The F- and R-symbols themselves are consistent with the SU(3) Clebsch–Gordan coefficients they are computed from: the F-move matches to 1e-12 and the R-move to 3e-15.

The multiplicity-free branch is unchanged, since R-symbols are scalars there.

🤖 Generated with Claude Code

[borisdevos]

Should wait till tests pass for certainty, but this change does make sense. Silly of me to not have caught this myself, thanks for the fix!

[codecov]

Codecov Report

✅ All modified and coverable lines are covered by tests.

Files with missing lines	Coverage Δ
src/sectors.jl	94.40% <100.00%> (ø)

... and 2 files with indirect coverage changes

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