Categorical representation theory

In $\operatorname{Rep}(S_3)$ over $\mathbb{C}$, with $V$ the $2$-dimensional irreducible representation, how many copies of $V$ occur in $V \otimes V \otimes V$?

What is $\operatorname{FPdim}(\operatorname{Rep}(D_4)) = \sum_i \operatorname{FPdim}(X_i)^2$, the sum over the simple objects of the representation category of the dihedral group of order $8$?

Which of these statements are true? (Choose all that apply.)

How many fusion categories over $\mathbb{C}$, up to monoidal equivalence, have Grothendieck ring $\Z[\Z/2]$ (two simple objects $\mathbf{1}, g$ with $g \otimes g \cong \mathbf{1}$)?

The proof that the Fibonacci fusion ring is not $K_0(\operatorname{Rep}(G))$ uses one property of $\operatorname{Rep}(G)$. Which?

The lesson's experiment finds, for the figure-eight knot, $\hat J = q^{-5} + q^{5}$, while $\mathit{Kh}$ has six one-dimensional summands in bidegrees $(t, q) = (-2, -5), (-1, -1), (0, -1), (0, 1), (1, 1), (2, 5)$. What does this show?

Which information about a fusion category $\mathcal{C}$ is lost when passing to its Grothendieck ring (as a ring with its basis of simple objects)? (Choose all that apply.)

Which Lean statement formalises “dimension is additive on short exact sequences $0 \to A \to B \to C \to 0$”? (Here f : A →ₗ[k] B, g : B →ₗ[k] C, and B is finite-dimensional.)

Write fusion_coefficients(sizes, chars) that takes the class sizes of a finite group and a list of its (real-valued) irreducible characters, each a list of values on the classes, and returns the nested list N[i][j][k] $= \langle \chi_i \chi_j, \chi_k \rangle = \frac{1}{|G|} \sum_c |c|\, \chi_i(c) \chi_j(c) \chi_k(c)$ as integers.