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Every self-dual simple object $X$ in a fusion category can canonically be assigned a number $a$, from its "snake" associator element:

snake associator

The square of $a$ equals Muger's "squared dimension" of $X$, an invariant of the fusion category. Let us work over $\mathbb{C}$. Then $a^2$ is positive real number, as shown by Etingof, Nikshych, and Ostrik.

The sign of $a$ is then well-defined and seems to also play an important role. It is also an invariant of the fusion category. For instance, in the Tambara-Yamagami categories, it equals the sign of $\tau$, an important invariant.

Are there other references where $\text{sign}(a)$ plays an important role? Is it studied in its own right somewhere?

Note that $\text{sign}(a)$ is closely related, but not equal to the 2nd Frobenius-Schur indicator $\nu_2(a)$ of $a$. The former is defined in a fusion category, depending only on the associators, while the latter requires a pivotal structure. For instance, in SuperVect, if $a$ is the odd one-dimensional vector space, then $\text{sign}(a) = 1$ but $\nu_2(a) = -1$.

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  • $\begingroup$ Did you try to insert a picture or formula of the snake associator? For some reason it's not there. Would you mind updating it? $\endgroup$ Aug 17, 2018 at 10:10

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