Grothendieck rings

The Grothendieck ring is a decategorification of a multiring category: objects are replaced by their classes, short exact sequences give additive relations, and tensor products give multiplication. In a semisimple category the additive relations are precisely the direct-sum relations. For a fusion category, the multiplication table is exactly the table of fusion rules.

Etingof et al. (2015), Chapter 3 develops the resulting combinatorics as the theory of $\mathbb Z_+$-rings; §§4.5 and 4.9 explain how these rings arise from tensor and fusion categories. TensorCategories.jl implements both passage from a category to its ring and direct computation with the resulting based ring.

Classes of objects

For an abelian category of finite-length objects, the Grothendieck group $\operatorname{Gr}(\mathcal C)$ has a generator $[X]$ for each isomorphism class, with relations

\[\label{eq:grothendieck-relation} [Y]=[X]+[Z] \quad\text{for every exact sequence }0\longrightarrow X\longrightarrow Y \longrightarrow Z\longrightarrow0.\]

Its basis consists of the classes of simple objects, and $[X]=\sum_i[X:S_i][S_i]$, where $[X:S_i]$ is a composition multiplicity (Etingof et al., 2015; Definition 1.5.8, p. 5).

When tensor product is exact in both variables, it induces multiplication $[X][Y]=[X\otimes Y]$, with identity $[\mathbb 1]$. Associativity follows from the associator, since isomorphic objects have equal classes (Etingof et al., 2015; §4.5, pp. 71–72).

The split Grothendieck ring instead imposes only the direct-sum relations $[X\oplus Y]=[X]+[Y]$. In a Krull–Schmidt category its basis consists of classes of indecomposable objects. For semisimple categories the two constructions agree, and the coefficients of $[X]$ are its simple-summand multiplicities.

The function split_grothendieck_ring(C) implements the direct-sum construction in models with finite indecomposable enumeration and tensor-product decomposition. Here “split” refers to the relations, not to a splitting field: the function also applies to supported non-split semisimple categories. For nonsemisimple input it does not impose relations from nonsplit exact sequences.

Positive bases and fusion rings

Write $b_i=[S_i]$. In the semisimple case the fusion rules are

\[\label{eq:fusion-ring-product} b_i b_j=\sum_l N_{ij}^{\,l}b_l,\qquad N_{ij}^{\,l}=[S_i\otimes S_j:S_l]\in\mathbb Z_{\geq0}.\]

Thus $\operatorname{Gr}(\mathcal C)$ is a ring over $\mathbb Z$ with a distinguished basis whose structure constants are nonnegative integers. A $\mathbb Z_+$-ring has such a basis and an identity whose coordinates in that basis are nonnegative; it is called unital when the identity itself is a basis element (Etingof et al., 2015; Definition 3.1.1, p. 49).

The positive cone $\sum_i\mathbb Z_{\geq0}b_i$ records actual objects of a semisimple category. The ring also contains virtual classes with negative coefficients. Its coefficient ring is $\mathbb Z$ even when the category is defined over a field of positive characteristic.

The package type is ZPlusRing; ℤ₊Ring and ℕRing are aliases for the same type. The last alias reflects nonnegative structure constants, but the mathematical terminology used in this manual is EGNO's $\mathbb Z_+$-ring.

For a split semisimple rigid category, duality induces a basis permutation $b_i^*=[S_i^*]$ and an anti-involution $(xy)^*=y^*x^*$. Let $I_0$ be the indices occurring in the unit and set $\tau(\sum_i a_ib_i)=\sum_{i\in I_0}a_i$. Then

\[\label{eq:based-ring-trace} \tau(b_i b_j)=\delta_{i,j^*}.\]

These are the based-ring conditions. A based ring of finite rank is a multifusion ring; it is a fusion ring when the unit is a basis element (Etingof et al., 2015; Definitions 3.1.3 and 3.1.7, pp. 49–50). In particular, the Grothendieck ring of a fusion category is a fusion ring (Etingof et al., 2015; Proposition 4.9.1, pp. 76–77).

With simple unit $b_u=1$, the last equation reads $N_{ij}^{\,u}=\delta_{i,j^*}$. A braiding makes the ring commutative, but commutativity alone does not specify a braiding.

Fusion matrices and Frobenius–Perron dimensions

For fixed $i$, the slice N[i,:,:] has entries

\[\label{eq:fusion-matrix} (M_i)_{j,l}=N_{ij}^{\,l}.\]

It is the matrix of left multiplication by $b_i$ in row coordinates: if $x$ has coefficient row $v$, then $b_ix$ has coefficient row $vM_i$.

For a fusion ring, the Frobenius–Perron dimension of $b_i$ is the spectral radius of $M_i$. Extending additively gives the unique ring homomorphism to $\mathbb R$ taking strictly positive values on the distinguished basis (Etingof et al., 2015; Definition 3.3.3 and Proposition 3.3.6, pp. 53–54). Consequently,

\[\label{eq:fpdim-multiplicativity} \operatorname{FPdim}(X\otimes Y) =\operatorname{FPdim}(X)\operatorname{FPdim}(Y).\]

These dimensions depend only on the fusion rules. They need not lie in the category's base field and do not require a pivotal structure. In contrast, dim(X) uses the chosen duality and pivotal data.

The package computes fpdim(r) from the integer multiplication matrices and returns an exact algebraic number. For a fusion ring $R$,

\[\label{eq:based-ring-fpdim} \operatorname{FPdim}(R)=\sum_i\operatorname{FPdim}(b_i)^2,\]

which is also fpdim(C) for a split fusion category with Grothendieck ring $R$.

Non-split categories

If $D_l=\operatorname{End}(S_l)$ is larger than the base field $k$, the multiplicity is

\[\label{eq:nonsplit-fusion-multiplicity} N_{ij}^{\,l} =\frac{\dim_k\operatorname{Hom}(S_l,S_i\otimes S_j)} {\dim_k D_l}.\]

This is the division used by multiplication_table(C) and coefficients(X,simples(C)) in the generic semisimple implementation.

Let $u$ denote the simple unit and put $D_u=\operatorname{End}(\mathbb 1)$. Rigidity gives

\[\label{eq:nonsplit-unit-coefficient} N_{ij}^{\,u}=\frac{\dim_kD_i}{\dim_kD_u}\,\delta_{i,j^*}.\]

Thus the coefficient of the unit in $[S_i][S_i^*]$ can exceed one. The resulting ring is a weak fusion ring (Etingof et al., 2015; §3.8, p. 63); see also Mäurer and Thiel (2024), §2.1.

For such a category the package uses

\[\label{eq:nonsplit-category-fpdim} \operatorname{FPdim}(\mathcal C) =\sum_i \frac{\dim_kD_u}{\dim_kD_i}\operatorname{FPdim}(S_i)^2.\]

Under the separability convention for categories over arbitrary fields, this is the categorical Frobenius–Perron dimension of Sanford (2025), Theorem 3.19 and Definition 3.21. When the unit is scalar, it is equation (5.3) of Mäurer and Thiel (2024), §5.3. The category method fpdim(C) includes these endomorphism-algebra factors for every supported semisimple input. The ring method fpdim(R) always sums the squares without these factors, so it does not give fpdim(C) in the non-split case.

What decategorification forgets

Decategorification retains tensor-product multiplicities but forgets the actual objects, morphisms, associator, and any additional monoidal structure. A categorification of a based ring is a category whose Grothendieck ring is the given based ring. A based ring can have inequivalent categorifications or none at all (Etingof et al., 2015; §§4.9–4.10).

For example, $\operatorname{Gr}(\operatorname{Vec}_k(G))=\mathbb Z[G]$, with one basis element for each group element. A 3-cocycle twist changes the associator but leaves this ring unchanged. For $\operatorname{Rep}_k(G)$, decategorification gives the representation ring, computed from tensor products of representations. These computations use the concrete categorical models introduced above.

An exact tensor functor induces a ring homomorphism $[X]\mapsto[F(X)]$. The converse is false: a homomorphism between Grothendieck rings does not determine a functor between the categories.

Continue with Computing with fusion rings for the package interface and examples, including a non-split representation ring.