Haagerup categories and data

The Haagerup subfactor has the smallest finite-depth index above $4$ Grossman and Snyder (2012), Introduction. Its Morita equivalence class contains exactly three fusion categories, denoted $H_1,H_2,H_3$ (Grossman and Snyder, 2012; Theorem 1.4). The constructors on this page include exact database records, a formula model, precomputed centers, additional stored tables, and fusion-ring-only entries. For each constructor, the entry states which structural data are supplied and which identification with the literature is known.

The default constructors load the following stored data:

The $H_1$ artifact

The loader haagerup_H1() returns a rank-four category over $\mathbb Q(\theta)$, where $\theta^4+\theta^3-\theta^2+\theta+1=0$, with labels $(\mathbb 1,\nu,\eta,\mu)$. The artifact supplies associators and all-one pivotal components, but no braiding or complex embedding. Its fusion table agrees with Mäurer (2026), §5.2, Table 2, p. 89, and with Barter et al. (2022), §V, Eq. (53), p. 13 after reordering that paper's $(\mathbb 1,\mu,\eta,\nu)$ to the package order. The artifact contains a separate exact quartic-field associator. The cited literature identifies the fusion ring, but it gives no change of fusion bases from this artifact gauge to the published unitary H1Data.m gauge.

The $H_2$ and $H_3$ loaders

The referenced $H_2$ and $H_3$ loaders use the labels $(\mathbb 1,\alpha,\alpha^*,\rho,\alpha\rho,\alpha^*\rho)$. They have the same fusion rules but different associators. Both AnyonWiki keys have braiding index zero, so these two loaders supply no $R$-symbols. Their common fusion ring can be written

\[\label{eq:haagerup-fusion-rules} \alpha^3=\mathbb 1,\qquad \alpha\rho=\rho\alpha^{-1},\qquad \rho^2=\mathbb 1+\rho+\alpha\rho+\alpha^2\rho.\]

Their identification is described in Mäurer et al. (2026), §1 and, with the keys and fusion rings printed explicitly, in Mäurer (2026), §5.2, Tables 2–3, pp. 89–91. Their literal $F$-symbol source is the corresponding (Vercleyen and Slingerland, Jun 2023) record, decoded as described on the AnyonWiki page; no entrywise change of fusion basis to the unitary formula gauges in the literature is recorded. The records also supply the dataset's all-one pivotal coefficients. The cited formula presentations do not identify these coefficients with their own pivotal gauges.

The $H_2$ entry stores the complex embedding that sends its displayed generator to approximately $0.651387818866+0.758744956776i$. The $H_3$ entry sends its generator to the real root approximately $1.722083805739$. The displayed generator $t$ is the archive generator. In the minimal-field notation of Mäurer et al. (2026), after Remark 4.13, p. 22, write $a=-t$ for $H_2$ and $b=-t$ for $H_3$. These generators are denoted $\alpha$ and $\beta$, respectively, in that reference; they are unrelated to the simple-object label $\alpha$ above. The substitution changes $t^4+t^3-t^2+t+1$ into $a^4-a^3-a^2-a+1$, and changes $t^4-t^3-t^2-t+1$ into $b^4+b^3-b^2+b+1$. This identifies the two presentations of the coefficient fields. The exact $F$-symbols remain identified by the pinned AnyonWiki record described above. The latter is the small-field exact presentation in Mäurer (2026), §5.2, p. 91, which is not in a unitary gauge; the Wolf constructor below is a separate unitary presentation. The call haagerup_H3(algebraic_closure(QQ)) extends the latter exact model to OSCAR's algebraic closure, choosing roots compatibly with this stored complex embedding. The overload haagerup_H3(:splitting_field) is currently unavailable; construct the algebraic closure explicitly as above.

using TensorCategories, Oscar
H = haagerup_H3()
@assert length(simples(H)) == 6
@assert H[2] ⊗ H[3] == one(H)
@assert H[4] ⊗ H[4] == H[1] ⊕ H[4] ⊕ H[5] ⊕ H[6]
@assert !is_braided(H)
base_ring(H)

Number field with defining polynomial x^4 - x^3 - x^2 - x + 1
  over rational field

Exact $H_3$ center archive

haagerup_H3_center() loads the split rank-twelve skeleton computed in Mäurer et al. (2026), §3.2, pp. 11–13 and Mäurer (2026), §5.2.2, pp. 93–94. The archive includes $F$-, $R$-, and pivotal data. It displays its simples as underlying $H_3$ objects equipped with half-braiding labels. In its stored order, these correspond to

\[\label{eq:haagerup-center-simple-order} (\mathbb 1,\pi_1,\pi_2,\sigma_0,\sigma_1,\sigma_2, \mu_1,\ldots,\mu_6).\]

The underlying objects of the first three entries are respectively $\mathbb 1$, $\mathbb 1\oplus\rho\oplus\alpha\rho\oplus\alpha^*\rho$, and $2\mathbb 1\oplus\rho\oplus\alpha\rho\oplus\alpha^*\rho$; those of the $\sigma_i$ are $\alpha\oplus\alpha^*\oplus\rho\oplus\alpha\rho\oplus\alpha^*\rho$, and those of the $\mu_i$ are $\rho\oplus\alpha\rho\oplus\alpha^*\rho$.

The coefficient field is $K_Z=\mathbb Q[t]/(p_Z(t))$, where the artifact records

\[\label{eq:haagerup-center-polynomial} \begin{aligned} p_Z(t)={}&t^{48}-t^{47}+2t^{46}-2t^{45}+2t^{44}-t^{43}-t^{42} +4t^{41}-8t^{40}+12t^{39}\\ &-15t^{38}+15t^{37}-10t^{36}+51t^{35}-31t^{34}+57t^{33} -27t^{32}+2t^{31}+59t^{30}\\ &-141t^{29}+229t^{28}-313t^{27}+342t^{26}-285t^{25}+85t^{24} +285t^{23}+342t^{22}\\ &+313t^{21}+229t^{20}+141t^{19}+59t^{18}-2t^{17}-27t^{16} -57t^{15}-31t^{14}-51t^{13}\\ &-10t^{12}-15t^{11}-15t^{10}-12t^9-8t^8-4t^7-t^6+t^5 +2t^4+2t^3+2t^2+t+1. \end{aligned}\]

If

\[\label{eq:haagerup-number-field} K_H=\mathbb Q[u]/(u^4-u^3-u^2-u+1)\]

is the defining field of haagerup_H3(), then $K_Z\cong K_H(\zeta_{39})$ (Mäurer, 2026; §5.2, Eqs. (5.5)–(5.6), p. 91). The polynomial $p_Z$ records the particular field generator used by the archive.

The loader chooses the complex embedding whose image of $t$ is the root nearest $1.29+0.25i$. The exact archive is not in a unitary gauge Mäurer et al. (2026), Remark 3.1, p. 13. This does not affect its pentagon and hexagon identities, but it distinguishes this presentation from the numerical unitary archive below.

Numerical center archive

The CSV files used by numeric_unitary_center_H3 are the published numerical data discussed in Mäurer (2026), §5.2.3, pp. 94–96. They were computed from the degree $16$ unitary Wolf presentation, beginning with $512$-bit balls; the source computation ended with a rigorous error bound of $107$ bits. The distributed CSV records decimal centers rather than the original ball radii. The loader parses those decimals into a new AcbField at the requested precision, so its output does not reproduce the original certifying balls. The files use the default :column_major_packing layout. In the thesis notation, their ten index columns are

\[\label{eq:haagerup-dictionary-index-order} (i,j,k,l,n,\delta,\gamma,m,\alpha,\beta),\]

whereas Mäurer (2026), §5.2.3.1 prints the mathematical projection labels as $(i,j,k,l,m,n,\beta,\alpha,\gamma,\delta)$. The default loader reads the published files in their stored column-major layout. The relation between the thesis's projection-inverse matrices and the package's structural matrices is given in $F$-symbol conventions.

The CSV archive does not contain pivotal components. The loader therefore uses the skeletal initializer's default coefficients $P_i=1$. Verify the pivotal or spherical identities at the chosen working precision before using dimensions, twists, or the $S$-matrix.

The numerical loader uses the same compact simple names $(\mathbb 1,\pi_1,\pi_2,\sigma_0,\sigma_1,\sigma_2,\mu_1,\ldots,\mu_6)$ as above. Unlike the exact archive, it stores no explicit half-braidings or underlying $H_3$ objects.

The optional precision argument is positional: for example, numeric_unitary_center_H3(96) requests $96$ bits. Requests above $107$ bits are capped because the source computation certifies only $107$ bits, even though the files print additional decimal digits. The constructor loads the files with the numerical loader's default check=false; it does not run the pentagon or hexagon checks automatically.

Wolf formulas and conventions

Osborne et al. (2019), Figure 1, Theorem 3.1, and Appendix B, pp. 2–3 and 16–23 supply a real solution with two signs $p_1,p_2\in\{\pm1\}$. The package contains a separate, unexported formula implementation TensorCategories.unitary_haagerup_H3_wolf. Its keyword defaults are p1=1 and p2=1.

The default coefficient argument constructs the degree $16$ number field defined by the polynomial in Osborne et al. (2019), Theorem 3.1. A supplied field must contain the required square roots. The constructor installs the fusion rules, tensor unit, and associators, uses the default pivotal coefficients $P_i=1$, and supplies no braiding.

With the embedding and radical choices that make the paper's radicals positive real, a matrix in the reference must be transposed to obtain the package's structural-matrix direction, as explained under $F$-symbol conventions. The formula file implements this translation. For each of the four sign pairs, every nonzero block agrees entrywise with Appendix B after this transpose, including the paper's channel order. The intended inputs satisfy $p_1,p_2\in\{\pm1\}$; the constructor does not validate this restriction or install the positive-real embedding.

Additional $H_2$ data

Unitary gauges make the adjoint operation transparent and are particularly useful for numerical and anyonic calculations. The following package artifact contains an additional exact associator table; its source does not record the corresponding fusion-basis gauge.

The exported constructor unitary_haagerup_H2() loads this separate exact associator table over the degree $8$ number field defined by $1+x^2-x^4+x^6+x^8$. The table is decoded using the default :column_major_packing layout, after which the constructor transposes every decoded matrix once more before storing it. The chosen embedding sends $x$ to approximately $-0.908677010512+0.417499809062i$. The constructor declares all-one pivotal components and supplies no braiding. It is not the data path used by haagerup_H2().

Despite its name, neither a source nor an explicit change of fusion bases identifies this table, including that extra transpose, with the unitary $H_2$ presentations in Barter et al. (2022), §V, pp. 13–14 or Huang and Lin (2020), §5.2.1, p. 16. No basis-level comparison with those presentations or independent pivotal-structure check is recorded.

Extended Haagerup

The Extended Haagerup subfactor is another exceptional finite-depth example. Its two even parts have ranks six and eight; the package records only the rank-six $M$–$M$ fusion ring constructed by Bigelow et al. (2012).

The extended-Haagerup helper TensorCategories.extended_haagerup(K) supplies the six labels

\[\label{eq:haagerup-izumi-simple-order} (\mathbb 1,f^{(2)},f^{(4)},f^{(6)},P',Q')\]

and the fusion rules of the $M$–$M$ even part in Bigelow et al. (2012), Appendix A, Table 13, p. 78. In the displayed order, the multiplication table in the implementation agrees entry by entry with that table. The $N$–$N$ even part has eight simple objects and is not returned by this constructor.

Only these fusion rules are supplied; extended-Haagerup associators are not implemented. In particular, the returned object records the Grothendieck-ring multiplicities but is not a verified monoidal category. Its identity associator blocks and all-one pivotal components are placeholders inherited from the skeletal initializer, not structural data for the extended-Haagerup category.