# Fusion category and induction matrix to its Drinfeld center: combinatorial properties

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.

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$$(\mathcal{C})^2 = 36$$.