Representations of finite groups
Finite-group representations are a fundamental source of symmetric tensor categories and the model behind ordinary finite-group symmetry. The constructor realizes $\operatorname{Rep}_K(G)$ as in Etingof et al. (2015), Examples 2.3.4 and 2.10.13, pp. 26 and 43: the tensor product uses the diagonal $G$-action, the unit is the trivial representation, and the symmetry is the ordinary flip. The package uses row coordinates, so action matrices and intertwiners multiply on the right. Explicitly, a stored row vector transforms by $v\mathbin{\cdot}g=v\rho(g)$, and an intertwiner $M:X\to Y$ satisfies $\rho_X(g)M=M\rho_Y(g)$. This right-action convention is equivalent to the usual left-action convention after replacing $g$ by $g^{-1}$.
representation_category(K,G) models finite-dimensional representations of a finite group over the specified field. Objects store a group homomorphism into a matrix group. Morphisms are intertwiners in the row-vector convention. Use either Representation(C,generators,matrices; check=false) for an existing category $C$, or Representation(G,generators,matrices; check=false) to infer the field and parent category from the matrices. The corresponding overloads with a Julia function evaluate that function on the package's generators.
Supplying action matrices
using TensorCategories, Oscar
G = cyclic_group(3)
C = representation_category(QQ,G)
A = matrix(QQ,[0 1; -1 -1])
X = Representation(C,gens(G),[A]; check=true)
@assert A^3 == identity_matrix(QQ,2)
@assert int_dim(End(X)) == 2
@assert is_simple(X)
f = morphism(X,X,A; check=true)
@assert matrix(f ∘ f) == A^2
int_dim(X ⊗ X)4The keyword defaults to check=false for both constructors. check=true on Representation checks the group relations, while on morphism it checks equivariance. Parent categories, endpoint dimensions, and coefficient fields are checked regardless. Supply a generating set of $G$ and one square action matrix for each generator, with a common size and coefficient field. A generator image may have smaller order than the generator.
Tensor products use diagonal group actions; duals use contragredient actions; the usual flip is symmetric in every characteristic. The canonical spherical structure is implemented. Tensor-product matrices use Kronecker products, with the coordinate from the right tensor factor varying fastest. No $F$-symbols are required.
Equality of representations compares their parent categories and their action matrices on the package's generators; representations related by a nontrivial change of basis are generally only isomorphic. Use is_isomorphic(X,Y) for that comparison. Two GroupRepresentationCategory values compare equal when their stored groups and coefficient fields compare equal.
Fields and enumeration
Maschke's theorem gives semisimplicity when the characteristic does not divide $|G|$. Splitting is a further condition. The rational example above is simple with a two-dimensional endomorphism field, so it is not absolutely simple. Finiteness of $G$ is a hypothesis of this model and is not checked by the constructor.
The no-field constructor representation_category(G) uses OSCAR's abelian closure of QQ. Even if the category is mathematically finite, simples(C) is not supported over every coefficient field. The current backend enumerates over finite fields (and handles the trivial group directly); it does not enumerate characteristic-zero irreducibles with their Schur-index information. Constructing explicit representations and computing their Hom spaces still works.
For small finite fields, the representation backend can enumerate simples and decompose modules. In modular characteristic distinguish composition factors from direct-sum summands. See Splitting.
Semisimplicity, splitting, and the distinction between simple and absolutely simple objects follow the conventions of Etingof et al. (2015), §§4.2 and 4.16.