Tambara–Yamagami and Ising categories
Tambara–Yamagami categories are the basic nonpointed extensions of the pointed category $\operatorname{Vec}_A$ by one simple object $m$. In the split setting, categories with these fusion rules are classified by a finite abelian group $A$, a nondegenerate symmetric bicharacter $\chi$, and $\tau\in k^\times$ satisfying $\tau^2=|A|^{-1}$ Tambara and Yamagami (1998), Theorem 3.2. The Ising fusion rules occur for $A=C_2$.
Let $A$ be a finite abelian group and $\chi:A\times A\to k^\times$ a nondegenerate symmetric bicharacter. The simple objects are the elements of $A$ and one object $m$, with
\[\label{eq:tambara-yamagami-fusion-rules} a\otimes b=ab,\quad a\otimes m=m=m\otimes a,\quad m\otimes m=\bigoplus_{a\in A}a.\]
The explicit normalization below is the one used in Gelaki et al. (2009), §4A, pp. 974–975.
The square-root parameter
The constructor tambara_yamagami(K,A,s,chi) takes $s^2=|A|$. In the cited paper the parameter satisfies $\tau^2=|A|^{-1}$. The translation is $\tau=1/s$. With the package's row-coordinate convention, the nontrivial blocks are
\[\label{eq:tambara-yamagami-associator} a_{a,m,b}=\chi(a,b)\mathrm{id}_m,\qquad a_{m,a,m}|_b=\chi(a,b)\mathrm{id}_b,\qquad (a_{m,m,m})_{a,b}=\frac{1}{s\chi(a,b)}.\]
The remaining blocks are identity blocks in the chosen skeletal ordering. The group simples are ordered by elements(A). Their printed names are the strings a1, a2, and so on, followed by m; these names do not print the group elements themselves.
These associator blocks agree exactly with the cited formulas after the translation $\tau=1/s$; no further gauge change is being suppressed. The constructor installs all-one pivotal components. With the package's chosen duality this makes the categorical dimension of $m$ equal to the supplied square root $s$, so its sign is part of the pivotal data. This pivotal normalization is an additional package choice rather than part of the cited monoidal formula comparison.
The field must contain the bicharacter values and the chosen nonzero square root. In particular, its characteristic cannot divide $|A|$. The constructor does not verify every bicharacter or parameter hypothesis: supplied data must satisfy them.
The available positional forms distinguish which data are supplied. In tambara_yamagami(K,A), both the bicharacter and $s$ are chosen in $K$; tambara_yamagami(K,A,s) fixes $s$ and chooses the bicharacter, while tambara_yamagami(K,A,chi) fixes the bicharacter and chooses $s$. The full form tambara_yamagami(K,A,s,chi) uses both supplied choices. The calls tambara_yamagami(A) and tambara_yamagami(n1,n2,...) make both automatic choices over a generated number field; the integer arguments are the abelian invariants used to construct $A$. The field-first integer form is tambara_yamagami(K,n1,n2,...). Without a field argument, the generated number field contains the required roots of unity and square root; the constructor does not use the algebraic closure itself.
Every automatic choice produces one category, rather than enumerating all Tambara–Yamagami categories for $A$. These general constructors supply no braiding. The Ising-specific constructor below separately attempts to add one.
For the trivial group, use the full call tambara_yamagami(K,A,s,chi) with an explicitly supplied $\chi$; the automatic bicharacter overload does not currently support this case.
using TensorCategories, Oscar
K, s = quadratic_field(2)
A = abelian_group(PcGroup,[2])
C = tambara_yamagami(K,A,s)
@assert length(simples(C)) == 3
@assert C[3] ⊗ C[3] == C[1] ⊕ C[2]
@assert C.ass[3,3,3,3] == inv(s)*matrix(K,[1 1; 1 -1])
@assert pentagon_axiom(C)Ising
The Ising category is the rank-three Tambara–Yamagami category for $A=C_2$. In physics its noninvertible simple object is the Ising or Majorana anyon, and the nontrivial invertible object is the fermion. The package uses labels $(\mathbb 1,\chi,X)$ for these three objects.
ising_category() uses $\mathbb Q(\sqrt2)$ and is unbraided because this field does not contain a primitive fourth root of unity. ising_category(K) chooses a square root of $2$ in $K$, while ising_category(K,s) uses the supplied root $s^2=2$; both use $q=1$ and attempt to install the corresponding braiding. The overload ising_category(K,q) instead chooses the square root and uses the specified integer $q$. The labels are $(\mathbb 1,\chi,X)$ in that order. Its associators are the Tambara–Yamagami formulas above for $A=C_2$ and $\chi(g,g)=-1$. The all-one stored pivotal components refer to the package's chosen duality, not to all dimensions being one.
ising_category(K,s,q) attempts to supply a braiding. To distinguish the program argument from the quadratic form in the reference, write $\varepsilon\in\{1,-1\}$ for the integer passed as q. Put $\xi=\varepsilon\zeta_4$, where root_of_unity(K,4) supplies $\zeta_4$, and choose $\alpha$ with $\alpha^2=(1+\xi)/s$. The implementation uses
\[\label{eq:ising-braiding} c_{\chi,\chi}=-1,\qquad c_{\chi,X}=c_{X,\chi}=\xi,\qquad c_{X,X}=\alpha\,\mathrm{id}_{\mathbb 1}\oplus \alpha\xi^{-1}\,\mathrm{id}_{\chi}.\]
This is the specialization of Galindo (2022), Theorem 4.9, Eq. (4.14), and Example 4.13 with $A=C_2$, $\tau=1/s$, and $Q(\chi)=\xi$; the scalar $\alpha$ is the same in both formulas. Thus the two summands of $c_{X,X}$ occur in the package's order $(\mathbb 1,\chi)$. If construction of the braiding data fails, for example because the field lacks the required roots, the constructor returns the underlying monoidal category. Check is_braided(C); absence of a braiding does not by itself diagnose the cause. The constructor does not check that the supplied integer is $\varepsilon=\pm1$, and an attempted installation is not itself a hexagon check. Supply one of the two stated values and verify hexagon_axiom(C) whenever braiding data are present. The choices of $\xi$ and $\alpha$ are part of these data.
L, z = cyclotomic_field(16)
I = ising_category(L)
@assert is_braided(I)
@assert pentagon_axiom(I) && hexagon_axiom(I)
@assert dim(I[3])^2 == 2
smatrix(I)[ 1 1 -z_16^6 + z_16^2]
[ 1 1 z_16^6 - z_16^2]
[-z_16^6 + z_16^2 z_16^6 - z_16^2 0]