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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.

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BQ1 has the following counter-example (inspired by a discussion with Corey Jones):

For $\mathcal{C} = Rep(D_9)$, the fusion rules are:

$$ \begin{smallmatrix}1&0&0&0&0&0\\ 0&1&0&0&0&0\\ 0&0&1&0&0&0\\ 0&0&0&1&0&0\\ 0&0&0&0&1&0\\ 0&0&0&0&0&1\end{smallmatrix} \ , \ \begin{smallmatrix}0&1&0&0&0&0\\ 1&0&0&0&0&0\\ 0&0&1&0&0&0\\ 0&0&0&1&0&0\\ 0&0&0&0&1&0\\ 0&0&0&0&0&1\end{smallmatrix} \ , \ \begin{smallmatrix}0&0&1&0&0&0\\ 0&0&1&0&0&0\\ 1&1&0&0&0&1\\ 0&0&0&1&1&0\\ 0&0&0&1&0&1\\ 0&0&1&0&1&0\end{smallmatrix} \ , \ \begin{smallmatrix}0&0&0&1&0&0\\ 0&0&0&1&0&0\\ 0&0&0&1&1&0\\ 1&1&1&0&0&0\\ 0&0&1&0&0&1\\ 0&0&0&0&1&1\end{smallmatrix} \ , \ \begin{smallmatrix}0&0&0&0&1&0\\ 0&0&0&0&1&0\\ 0&0&0&1&0&1\\ 0&0&1&0&0&1\\ 1&1&0&0&1&0\\ 0&0&1&1&0&0\end{smallmatrix} \ , \ \begin{smallmatrix}0&0&0&0&0&1\\ 0&0&0&0&0&1\\ 0&0&1&0&1&0\\ 0&0&0&0&1&1\\ 0&0&1&1&0&0\\ 1&1&0&1&0&0\end{smallmatrix} $$ and the induction matrix is: $$ \left(\begin{smallmatrix} 1&0&1&1&1&1&0&0&0&0&0&0&0&0&0&0&0&0&0&0&0&0&0&0&0&0&0&0&0&0&0&0&0&0&0&0&0&0&0&0&0&0&1&0\\ 0&1&1&1&1&1&0&0&0&0&0&0&0&0&0&0&0&0&0&0&0&0&0&0&0&0&0&0&0&0&0&0&0&0&0&0&0&0&0&0&0&0&0&1\\ 0&0&0&0&0&0&1&1&1&1&1&1&1&1&1&0&0&0&0&0&0&0&0&0&0&0&0&0&0&0&0&0&0&0&0&0&0&0&0&0&0&0&1&1\\ 0&0&0&0&0&0&0&0&0&0&0&0&0&0&0&1&1&1&1&1&1&1&1&1&0&0&0&0&0&0&0&0&0&0&0&0&0&0&0&0&0&0&1&1\\ 0&0&0&0&0&0&0&0&0&0&0&0&0&0&0&0&0&0&0&0&0&0&0&0&1&1&1&1&1&1&1&1&1&0&0&0&0&0&0&0&0&0&1&1\\ 0&0&0&0&0&0&0&0&0&0&0&0&0&0&0&0&0&0&0&0&0&0&0&0&0&0&0&0&0&0&0&0&0&1&1&1&1&1&1&1&1&1&1&1 \end{smallmatrix}\right) $$ It follows that $44=$ rank$(Z(\mathcal{C})) \not \le$ rank$(\mathcal{C})^2 = 36$.

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