The Centralizer Construction

Let $\mathcal C$ be a tensor category and $\mathcal D \subset \mathcal C$ a full topologizing subcategory. Then the centralizer $\mathcal Z(\mathcal C \colon \mathcal D)$ is given by tuples $(Z,\gamma)$ where $\{\gamma_(X)\colon Z\otimes X \to X \otimes Z \mid X \in \mathcal D}$ satisfies the hexagon equations.

Example

Let $G = S_3$. We want to compute the centralizer of $\langle \delta_{(1,2)}\rangle$.

G = symmetric_group(3)

Vec = graded_vector_spaces(QQ,G)

C = centralizer(Vec, Vec[2])

simples(C)

6-element Vector{TensorCategories.CentralizerObject}:
 Central object: Graded vector space of dimension 1 with grading
PermGroupElem[()]
 Central object: Graded vector space of dimension 1 with grading
PermGroupElem[()]
 Central object: Graded vector space of dimension 1 with grading
PermGroupElem[(2,3)]
 Central object: Graded vector space of dimension 1 with grading
PermGroupElem[(2,3)]
 Central object: Graded vector space of dimension 2 with grading
PermGroupElem[(1,3), (1,2)]
 Central object: Graded vector space of dimension 2 with grading
PermGroupElem[(1,3,2), (1,2,3)]