This question is inspired by this paper of Scott Morrison and Kevin Walker.
Consider a fusion category $\mathcal{C}$ of rank $r$, and its Drinfeld center $Z(\mathcal{C})$ of rank $s$.
Let $N_i = (n_{ij}^k)$, $1 \le i \le r $, be the $r\times r$ fusion matrices of $\mathcal{C}$.
Let $I_{\mathcal{C}} = (m_{kl})$ be the $r \times s$ matrix for the induction from $\mathcal{C}$ to $Z(\mathcal{C})$.
Consider $c(\mathcal{C},k,a) = card(\{ (i,j) \ | \ n_{ij}^k = a \})$ and $c(I_{\mathcal{C}},k,a) = card(\{ l \ | \ m_{kl} = a \})$.
Question 1: Is the following equality true for $\mathcal{C} = Rep(G)$ with $G$ a finite group, and $a>0$? $$c(\mathcal{C},k,a) = c(I_{\mathcal{C}},k,a)$$
It is checked for $G = C_2, C_3, S_3, D_5, A_5$. We also found formal solutions for $I_{\mathcal{C}}$ with the above equality for $G= A_4, D_4, Q_8$. So is for the Hopf ${\rm C^*}$-algebras of dimensions $8$ and $12$. Then:
Question 2: It is also true for $\mathcal{C} = Rep(\mathbb{A})$ with $\mathbb{A}$ a (finite dimensional) Hopf ${\rm C^*}$-algebra?
Note that it is false for the Extended-Haagerup fusion categories $EH_1$ and $EH_2$.
We can check Q1 directly for $G=S_3$ and $A_5$ by observing the matrices below.
For $\mathcal{C} = Rep(S_3)$, the fusion matrices are $$ \left(\begin{matrix} 1 & 0 & 0 \\ 0 & 1 & 0 \\ 0 & 0 & 1 \end{matrix} \right), \left(\begin{matrix} 0 & 1 & 0 \\ 1 & 0 & 0 \\ 0 & 0 & 1 \end{matrix} \right), \left(\begin{matrix} 0 & 0 & 1 \\ 0 & 0 & 1 \\ 1 & 1 & 1 \end{matrix} \right), $$ and the induction matrix is
$$ \left(\begin{matrix} 0 & 1 & 0 & 0 & 0 & 1 & 0 & 1 \\ 1 & 0 & 0 & 0 & 0 & 1 & 1 & 0\\ 1 & 1 & 1 & 1 & 1 & 0 & 0 & 0 \end{matrix} \right). $$
For $\mathcal{C} = Rep(A_5)$, the fusion matrices are
$$ \left(\begin{smallmatrix}1&0&0&0&0\\0&1&0&0&0\\0&0&1&0&0\\0&0&0&1&0\\0&0&0&0&1 \end{smallmatrix} \right), \left(\begin{smallmatrix}0&1&0&0&0\\1&1&0&0&1\\0&0&0&1&1\\0&0&1&1&1\\0&1&1&1&1 \end{smallmatrix} \right), \left(\begin{smallmatrix}0&0&1&0&0\\0&0&0&1&1\\1&0&1&0&1\\0&1&0&1&1\\0&1&1&1&1 \end{smallmatrix} \right), \left(\begin{smallmatrix}0&0&0&1&0\\0&0&1&1&1\\0&1&0&1&1\\1&1&1&1&1\\0&1&1&1&2 \end{smallmatrix} \right), \left(\begin{smallmatrix}0&0&0&0&1\\0&1&1&1&1\\0&1&1&1&1\\0&1&1&1&2\\1&1&1&2&2 \end{smallmatrix} \right), $$ and the induction matrix is
$$ \left(\begin{matrix} 0&0&1&1&0&0&0&0&0&0&0&0&0&0&0&1&1&0&0&0&0&1 \\ 1&1&0&1&1&1&1&0&0&0&0&1&1&1&1&1&1&0&0&0&1&0 \\ 1&1&0&1&1&1&1&1&1&1&1&0&0&0&0&1&1&0&0&1&0&0 \\ 1&1&1&2&1&1&1&1&1&1&1&1&1&1&1&0&0&0&1&0&0&0 \\ 2&2&2&1&1&1&1&1&1&1&1&1&1&1&1&1&1&1&0&0&0&0 \end{matrix} \right). $$
The above matrices come from personal computation and should be confirmed.
Any other combinatorial properties (proved or asked) are welcome. For examples:
Bonus question 1: Is it true that rank$(Z(\mathcal{C})) \le $ rank$(\mathcal{C})^2$? If so, is there a better bound?
Let $m_1$ and $m_2$ be the maximal entry of the fusion matrices of $\mathcal{C}$, and of the matrix $I_{\mathcal{C}}$, respectively.
Bonus question 2: Is it true that $m_1 \ge m_2$.
BQ1 and BQ2 are checked by all the fusion categories cited above.
