Vercleyen–Slingerland imports

These two examples arise from the “song” construction of Vercleyen and Slingerland (2023), §5.1, Examples (3)–(4), pp. 10–11, which generalizes Tambara–Yamagami and Haagerup–Izumi fusion rings. The rank-eight ring is obtained from a crossed product of a Tambara–Yamagami category. The rank-nine ring is categorifiable, but it is not of Tambara–Yamagami or Haagerup–Izumi type and does not belong to the crossed-product families considered there.

These constructors load bundled associator files accompanying the published fusion rings. The files do not record enough basis metadata to translate their matrix entries to a published $F$-symbol convention. Run pentagon_axiom when coherence of a selected import is required.

cat_fr_8122(n) has the fusion ring

\[\label{eq:vercleyen-slingerland-rank-eight-label} FR^{8,1,2}_2= \left[\mathbb Z_3\mathrel{\trianglelefteq}D_3\right]^{\mathrm{Id}}_{1\mid0},\]

which is entry 155 in Example 5.1(3) of the cited paper. Its multiplication table agrees with that description. The source directory is named FR_8211, but that name is a typo and is not the mathematical identifier. The constructor loads the $n$-th of 96 bundled files corresponding to the paper's ancillary file CategorificationsFR_8_1_2_2.wdx. Its coefficient field is $\mathbb Q(\zeta_{24})$ and its label order is $(e,a,b,aba,t,s,ba,ab)$. It installs the imported associator blocks and retains the skeletal constructor's default all-one pivotal components, without validating those components against the associators. It supplies no braiding. The argument must satisfy $1\leq n\leq96$.

cat_fr_9143() has the fusion ring

\[\label{eq:vercleyen-slingerland-rank-nine-label} FR^{9,1,4}_3= \left[\mathbb Z_2\mathrel{\trianglelefteq}\mathbb Z_6\right]^\alpha_{1\mid0}, \qquad \alpha(g)=g^{-1},\]

which is entry 305 in Example 5.1(4). Its multiplication table likewise agrees with the published ring. The constructor loads the data corresponding to the ancillary file CategorificationsFR_9_1_4_3.wdx over OSCAR's algebraic closure QQBarField(), with labels $(g_0,g_3,t_2,t_1,t_0,g_4,g_2,g_5,g_1)$. It installs fusion rules, associator blocks, a tensor unit, and the default pivotal coefficients $P_i=1$, but no braiding.

The paper and its ancillary files identify the fusion rings and the source of the coefficients, but they do not specify the row and column bases used by the repository import. The small-block ordering is therefore not identified, and the imported ordering differs from the AnyonWiki dictionary format. These constructors therefore provide the published fusion rings and bundled associator matrices without a basis-level identification with the cited $F$-symbol presentations.