If I consider a simple object $X$ in a fusion category and tensor it with its dual $X^\*$, and let $Y$ be a simple object in the decomposition $X \otimes X^\* = I + Y + \dotsb$. I want to say that $Y$ cannot be pseudo-real, i.e. $Y^{\*\*} = -Y$ (Frobenius-Schur indicator $= -1$), because $(X \otimes X^\*)^{\*\*} = (X^{\*\*} \otimes X^\*)^\* = X^{\*\*} \otimes X^{\*\*\*}$, and should equal to $X \otimes X^\*$. Is this correct?


Perhaps the term "pseudo-real" is not standard in fusion category, but in the case of a $\operatorname{Rep}(G)$ category for some group $G$, my question above becomes the following. Can a pseudo-real representation appear in the decomposition of the tensor product of an irreducible representation $\mathbf r$ with its conjugate representation $\bar{\mathbf r}$?



Pseudo-real (Frobenius-Schur indicator $= -1$) simple object in $X \otimes X^*$?
If $Y$ appears with multiplicity one, then it cannot be pseudo-real. The proof is that it has a symmetric pairing given by restricting the symmetric pairing on $X \otimes X^\*$.


However, if $Y$ appears with even multiplicity, then there are counterexamples, for example see [this answer](https://mathoverflow.net/a/55420/22).


