What do "pivotal" and "spherical" mean for (unitary) fusion categories on the level of the $F$-symbols?

Question

For me, a fusion category (over $\mathbb{C}$) is just a tensor $F$ (the associator, with $6$ simple-object labels and $4$ fusion space indices) and a tensor $d$ (the quantum dimensions, with one simple-object label), such that the $F$-tensor fulfils the pentagon equation. (This seems to be the most useful language for physics.)

If furthermore the fusion category is unitary, this means that (one can find a basis for the fusion spaces such that) the $F$-tensor is unitary when interpreted as a linear map between two groups of its indices.

What extra structure/conditions do I have to add in this tensor language to make the (unitary) fusion category pivotal/spherical?

For example, it seems that a unitary fusion category is spherical if the $F$-tensor is symmetric under the full tetrahedral symmetry group (after normalizing with the square roots of the quantum dimensions of four of its simple-object labels). This seems not to be the most general case, however.

Answer 1

Any unitary fusion category has a canonical spherical structure. See Example 2.12 of this paper and the references therein. It also follows from a more general result Prop 8.23 in this paper. So you don't need to do anything in the unitary case.

For a non-unitary fusion category, I think you're looking for Bruce Bartlett's explicit description of pivotal structures. There he shows that you can think of a pivotal structure as a choice of a number $\gamma_i$ attached to each simple object such that the condition of Theorem 5.4 holds: $$\gamma_j \gamma_k \mathrm{Id} = \gamma_i T_{jk}^i \text{ whenever $X_i$ is a summand of $X_j \otimes X_k$}.$$ Here $T_{jk}^i$ are the pivotal operators which are defined from the F-matrices by Thm 4.17. (There's a slightly tricky point there which I'm unclear on, which is that the formula in 4.17 only works for a "fair basis" which should translate into some simple condition on how you've gauaged your F-matrix, but it's not obvious to me what that condition is.) The pivotal structure is spherical if all the $\gamma_i$ are $\pm 1$.

Answer 2

This is really an answer to the question in the comment of Noah's answer.

Let $F$ be a solution to the pentagon equations for some fusion category $\mathcal C$. Pivotal structures on $\mathcal C$ should be in $1-1$ correspondence with solutions to the polynomial equations

\begin{equation} \epsilon_c^{-1} \epsilon_b \epsilon_a = \sum_{s=1}^{N_{b c^*}^{a^*}}\sum_{t=1}^{N_{c^* a}^{b^*}} F_{abc^*}^{\mathbf 1}\lbrack\begin{smallmatrix}i & c & 1 \\ 1 & a^* & s\end{smallmatrix}\rbrack F_{bc^*a}^{\mathbf 1}\lbrack\begin{smallmatrix}s & a^* & 1 \\ 1 & b^* & t\end{smallmatrix}\rbrack F_{c^* ab}^{\mathbf 1}\lbrack\begin{smallmatrix}t & b^* & 1 \\ 1 & c & i\end{smallmatrix}\rbrack \end{equation}

for all $i \in \lbrace 1,\ldots, N_{ab}^c\rbrace$, where $N_{ab}^c$ is the dimension of $hom(a \otimes b,c)$. The $\epsilon_a$ are referred to as pivotal coefficients. There are various ways in which the pivotal coefficients and quantum dimensions for a given solution can be played around with, given our initial solution $F$ to the pentagon equations. For ease, we'll go ahead and assume that our $F$ matrices use basis choices which agree with those in 1305.2229 (which are pretty standard choices), in which case the left and right quantum dimensions of an object $a$ are related to $(F,\epsilon)$ via

\begin{align} q_l(a) &= \epsilon_a\left ( F_{a^*a a^*}^{a^*}\lbrack\begin{smallmatrix}1 & 1 & 1 \\ 1 & 1 & 1\end{smallmatrix}\rbrack\right)^{-1}, & q_r(a) &= \left (\epsilon_a F_{a a^* a}^{a}\lbrack\begin{smallmatrix}1 & 1 & 1 \\ 1 & 1 & 1\end{smallmatrix}\rbrack\right)^{-1}. \end{align}