Let $G$ be a finite group and $\pi$ an irreducible complex representation. The Frobenius-Schur indicator of $\pi$ is defined as:

$$ \nu_2(\pi):=\frac{1}{|G|} \sum_{g \in G} \chi_{\pi}(g^2) $$
with $\chi_{\pi}$ the character of $\pi$. Recall that $\nu_2(\pi) \in \{-1, 0,1\}$. The only finite simple groups $G$ with $|G|<10^7$ and having an irreducible complex representation $\pi$ with $\nu_2(\pi) = -1$ are $\mathrm{PSU}(3,q)$ with $q=3,4,5,7,8$ and $\mathrm{PSU}(4,3)$.

```
gap> it:=SimpleGroupsIterator(10,10000000);; for g in it do if -1 in Indicator(CharacterTable(g),2) then Print([g]); fi; od;
[ PSU(3,3)][ PSU(3,4)][ PSU(3,5)][ PSU(4,3)][ PSU(3,8)][ PSU(3,7)]
gap>
```

**Question**: What are the finite simple groups known to have no irreducible complex representation $\pi$ with $\nu_2(\pi) = -1$?

What are those known to have an irreducible complex representation $\pi$ with $\nu_2(\pi) = -1$?

I am specifically interested in the groups $\mathrm{PSL}(2,q)$ for which I ckecked by GAP that for $q<500$, there is no irreducible complex representation $\pi$ with $\nu_2(\pi) = -1$:

```
gap> for q in [2..500] do if -1 in Indicator(CharacterTable( "PSL", 2, q),2) then Print([q]); fi; od;
gap>
```

So I expect that it is true for all $q$. Is there a proof using the character table (including class type) of $\mathrm{PSL}(2,q)$ (see this post)?