Concrete models: graded spaces and representations

The matrix tutorial can be extended by adding structure to each vector space and restricting the allowed matrices. This is how the package implements graded vector spaces and group representations. It need not first decompose every tensor product into simples or compute any $F$-symbols.

In both models, forgetting the additional structure gives a faithful linear realization

\[\label{eq:underlying-vector-space-functor} U:\mathcal C\longrightarrow\operatorname{Vec}_k,\]

and the stored matrix of a morphism represents its image under $U$. For $\operatorname{Rep}_k(G)$ and untwisted graded vector spaces this is the usual fiber functor. For a cocycle-twisted graded category it remains a faithful linear realization, but a nontrivial cohomology class can obstruct compatible tensorators. Thus concrete matrix coordinates do not by themselves require a fiber functor; see Fiber functors and semisimple coordinates.

Graded vector spaces

For a finite group $G$, the category of finite-dimensional $G$-graded vector spaces, with associator twisted by a normalized $3$-cocycle $\omega$, is usually denoted $\operatorname{Vec}_G^\omega$. Store a vector space together with the degree of each basis vector. A morphism matrix may have a nonzero entry only between basis vectors of the same degree. Give a tensor basis vector $v\otimes w$ degree $\deg(v)\deg(w)$, in this order even for nonabelian $G$. Order the tensor basis so that the coordinate of $w$, the right factor, varies fastest; then tensor products of morphisms are the corresponding Kronecker products.

The existing representation uses GVSObject with fields V and grading, and GVSMorphism with a matrix. Here is a computation with those types:

using TensorCategories, Oscar
G = cyclic_group(3)
g = first(gens(G))
C = graded_vector_spaces(QQ, G)
X = C[one(G), g]
Y = C[g]
f = morphism(X, Y, matrix(QQ, 2, 1, [0, 1]))
@assert int_dim(Hom(X,Y)) == 1
@assert int_dim(X ⊗ Y) == 2
K, inclusion = kernel(f)
@assert int_dim(K) == 1 && is_zero(f ∘ inclusion)
matrix(f)

[0]
[1]

For the untwisted category, the associator is the canonical rebracketing map in the chosen tensor bases. For a normalized $3$-cocycle $\omega$, modify the associator on homogeneous tensors by $\omega(g,h,l)$. This is additional data, and its cocycle equation is precisely the relevant coherence condition (Etingof et al., 2015; §2.3). A dual basis vector to one of degree $g$ has degree $g^{-1}$; in the twisted case the evaluation normalization also depends on $\omega$. A braiding requires compatible extra data and, in particular, is not supplied by an arbitrary group grading or $3$-cocycle. The current braiding method is available only for an untwisted category with a stored bilinear form: the ordinary constructor stores the trivial form when $G$ is abelian, and the bilinear-form constructor permits other choices. See the graded-vector-space catalogue entry for the exact implemented constructors and limitations.

Representations

The same pattern implements $\operatorname{Rep}_k(G)$. Store the action of generators of $G$ on each representation. A matrix $M$ from $X$ to $Y$ is a morphism when

\[\label{eq:representation-intertwiner} \rho_X(g)M=M\rho_Y(g)\]

for all generators. This is the row-vector convention. Solving these linear equations gives a Hom basis. Direct sums use block actions and tensor products use Kronecker products, again with the right-factor coordinate varying fastest. Kernels of intertwiners are invariant subspaces; to return a kernel object, restrict the group action to a basis of that subspace. The dual has the contragredient action $\rho^*(g)=\rho(g^{-1})^{\mathsf T}$ in these row coordinates, and the ordinary flip gives the symmetric braiding. These structures are implemented; see the representation catalogue entry.

R = representation_category(QQ, G)
A = matrix(QQ, [0 1; -1 -1])
V = Representation(R, gens(G), [A]; check=true)
@assert A^3 == identity_matrix(QQ,2)
@assert int_dim(End(V)) == 2
@assert is_simple(V)
int_dim(V ⊗ V)

This is a simple rational representation whose endomorphism algebra is isomorphic to $\mathbb Q(\zeta_3)$. No $F$-symbol data were used. Enumeration of all rational irreducibles is a separate backend capability, not a prerequisite for constructing this object or computing its Hom spaces.

These remain concrete category models: their objects and morphisms retain the underlying graded spaces or representations. They do not need to be converted to $F$-symbol data. If a split fusion category supports the necessary decomposition algorithms, the later section on extracting a skeleton explains how to choose fusion bases and pass to such coordinates. Non-split and nonsemisimple concrete models remain usable even when that scalar $F$-symbol construction does not apply.

Continue with Skeletal fusion categories, where a split fusion category is represented by simple multiplicities and structural matrices rather than by underlying vectors or group actions.