Let $\mathcal{C}$ be a complex fusion category. If it admits a pivotal structure $a$ then by [1, Proposition 4.7.12], $\dim_a$ induces a character $\chi$ on the Grothendieck ring $Gr(\mathcal{C})$, of basis say $(b_i)$. As a ring homomorphism, $\chi$ corresponds to a one-dimensional representation $E_{\chi}$ of $Gr(\mathcal{C})$. Then the categorical dimension of $\mathcal{C}$ is $\sum_i |\chi(b_i)|^2$ which, by [2, Example 2.4], is precisely the formal codegree $f_{E_{\chi}}$.
Conclusion, in the pivotal case, the categorical dimension must be the formal codegree of a one-dimensional representation of $Gr(\mathcal{C})$.
Question: In general (i.e. without assuming pivotal), should the categorical dimension be a formal codegree (of a non-necessarily one-dimensional representation)?
Note that the existence of a pivotal structure on every semisimple tensor category (in particular, fusion category) is a well-known open problem [1, Question 4.8.3]. If it is true then so is our question (trivially). But our question could have been studied independently.
References
[1]: P. Etingof, S. Gelaki, D. Nikshych, V. Ostrik; Tensor categories. Mathematical Surveys and Monographs, 205.
[2]: V. Ostrik, On formal codegrees of fusion categories. Math. Res. Lett. 16 (2009), no. 5, 895–901.
