Computing with fusion rings

This page applies the preceding Grothendieck-ring conventions. It first constructs the ring of a category, then enters fusion rules without a categorification, and finally compares a non-split category with its splitting field.

A split representation ring

Over $\mathbb F_3$, the cyclic group $C_2$ has the trivial and sign representations. Their classes satisfy $[\varepsilon]^2=[\mathbb 1]$.

using TensorCategories, Oscar
G = cyclic_group(2)
C = representation_category(GF(3), G)
S = simples(C)
R = split_grothendieck_ring(C)
unit_index = only(findall(X -> is_isomorphic(X, one(C))[1], S))
sign_index = only(setdiff(eachindex(S), [unit_index]))
u, e = R[unit_index], R[sign_index]
@assert u == one(R) && e*e == u
@assert involution(e) == e
e*e

X2

The basis of R follows the simple-object order of C. To obtain the class of an object, pass its integer multiplicities to R:

Y = S[sign_index] ⊕ S[sign_index]
y = R(ZZ.(coefficients(Y,S)))
@assert y == 2*e
coefficients(y)

2-element Vector{ZZRingElem}:
 2
 0

Here ZZ.(...) converts each multiplicity to an OSCAR integer. A virtual class such as e-u is also an element of R:

@assert base_ring(R) == ZZ
@assert fpdim(e) == 1
@assert fpdim(R) == 2
fpdim.(basis(R))

2-element Vector{QQBarFieldElem}:
 {a1: 1.00000}
 {a1: 1.00000}

Both Frobenius–Perron dimensions are $1$, and their squared sum is $2$. The main ring operations are:

For semisimple rigid input, split_grothendieck_ring stores the involution obtained from the duality permutation.

Entering a ring without a category

ZPlusRing constructs a ring directly from its basis names, multiplication table, and unit coefficient vector. Its aliases are ℤ₊Ring and ℕRing. For the Fibonacci rule $t^2=1+t$:

using TensorCategories, Oscar
N = zeros(Int,2,2,2)
N[1,1,1] = N[1,2,2] = N[2,1,2] = 1
N[2,2,1] = N[2,2,2] = 1
R = ZPlusRing(["1","t"], N, [1,0])
t = R[2]
@assert t*t == one(R)+t
d = fpdim(t)
@assert d > 0 && d^2 == 1+d
d

{a2: 1.61803}

Thus $\operatorname{FPdim}(t)=(1+\sqrt5)/2$. No associator or coefficient field for a categorification was needed. Constructing a category with this Grothendieck ring requires substantially more data. The constructor converts the supplied table and unit coordinates to integers; it does not itself certify nonnegativity, associativity, the unit equations, or the based-ring identities. These are assumptions on directly entered data.

A non-split representation ring

Over $\mathbb F_2$, the group $C_3$ has two irreducible representations: the trivial representation and a two-dimensional representation $V$ with endomorphism field $\mathbb F_4$. The category is semisimple, and

\[\label{eq:rank-two-fusion-rule} [V]^2=2[\mathbb 1]+[V].\]

Indeed, over $\mathbb F_4$ the representation $V$ splits as $\chi\oplus\chi^{-1}$, so its square is $2\cdot\mathbb 1\oplus\chi\oplus\chi^{-1}$.

using TensorCategories, Oscar
G = cyclic_group(3)
C = representation_category(GF(2),G)
S = simples(C)
R = split_grothendieck_ring(C)
i = only(findall(Y -> int_dim(Y) == 2,S))
V, v = S[i], R[i]
@assert int_dim(End(V)) == 2
@assert v*v == 2*one(R)+v
@assert fpdim(v) == 2
@assert fpdim(C) == 3
@assert fpdim(R) == 5
v*v

2⋅X1 + X2

The coefficient $2$ is an integer multiplicity; it does not vanish in the Grothendieck ring. The category's Frobenius–Perron dimension is $1^2+2^2/2=3$, whereas the ring method's unweighted sum is $1^2+2^2=5$.

Over the splitting field, there are three one-dimensional simples:

Cs = representation_category(GF(2,2),G)
Rs = split_grothendieck_ring(Cs)
@assert rank(Rs) == 3
@assert all(b -> fpdim(b) == 1,basis(Rs))
@assert fpdim(Rs) == 3
rank(Rs)

3

The field extension changes the simple basis and hence the Grothendieck ring; its multiplication coefficients are integers over both fields.

Continue with pivotal and spherical structures.