The $E_6$ constructor

The half-$E_6$ category is the rank-three fusion category formed by the even sectors of the $E_6$ subfactor. Its Drinfeld center is one of the exotic modular categories studied by Hong et al. (2008), §2.1, pp. 1053–1054, whose §8.1 also records explicit recoupling data for the half-$E_6$ category. The package entry is intended to encode that presentation, but the current coefficients do not do so correctly.

E6subfactor() uses OSCAR's algebraic closure QQBarField() and labels $(\mathbb 1,x,y)$, with

\[\label{eq:e6-fusion-rules} x^2=\mathbb 1+2x+y,\qquad xy=yx=x,\qquad y^2=\mathbb 1.\]

The implementation contains explicit blocks of sizes $1$, $2$, and $6$, including complex roots of unity. Its source points to Hong et al. (2008) for the recoupling data.

In particular, the $6\times6$ block uses $\zeta_{24}^{-3}$ in entries where Hong et al. (2008), §8.1, p. 1069 print $\exp(-\pi i/3)=\zeta_{24}^{-4}$. These roots have different orders and cannot be exchanged by a cyclotomic-field automorphism. That reference orders its channels as $\mathbb 1$, $y$, then the four $x$ channels. The package orders the same block as $\mathbb 1$, the four paths through $x$, then $y$, following the general path ordering under $F$-symbol conventions. This discrepancy is only one obstruction: changing that scalar alone does not make the stored table satisfy the pentagon. No braiding is supplied, and the output uses the skeletal initializer's default coefficients $P_i=1$. Since the associator is not coherent, these components do not define a pivotal structure on a monoidal category. The output must not be used as the half-$E_6$ fusion category.