The complex irreps of a finite group come in three types: self-dual by a
symmetric form, self-dual by a symplectic form, and not self-dual at all.
In the first two cases, the character is real-valued, and in the third
it is sometimes only complex-valued. The cases can be distinguished by
the value of the Schur indicator $\frac{1}{|G|} \sum_g \chi(g^2)$,
necessarily $1$, $-1$, or $0$. They correspond to the cases that
the representation is the complexification of a real one, the forgetful
version of a quaternionic representation, or neither.
A conjugacy class $[g]$ is called "real" if all characters take real
values on it, or equivalently, if $g\sim g^{-1}$. I vaguely recall the
number of real conjugacy classes being equal to the number of real irreps.
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Do I remember that correctly?
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Can one split the real conjugacy classes into two types,
"symmetric" vs. "symplectic"?
With #1 now granted, a criterion for a "good answer" would be that the number of symmetric real conjugacy classes should equal the number of symmetrically self-dual irreps.
(I don't have any application in mind; it's just bothered me off and on
for a long time.)
Best Answer
It's a great question! Disappointingly, I think the answer to (2) is No :
The only restriction on a `good' division into "symmetric" vs. "symplectic" conjugacy classes that I can see is that it should be intrinsic, depending only on $G$ and the class up to isomorphism. (You don't just want to split the self-dual classes randomly, right?) This means that the division must be preserved by all outer automorphisms of $G$, and this is what I'll use to construct a counterexample. Let me know if I got this wrong.
The group
My $G$ is $C_{11}\rtimes (C_4\times C_2\times C_2)$, with $C_2\times C_2\times C_2$ acting trivially on $C_{11}=\langle x\rangle$, and the generator of $C_4$ acting by $x\mapsto x^{-1}$. In Magma, this is G:=SmallGroup(176,35), and it has a huge group of outer automorphisms $C_5\times((C_2\times C_2\times C_2)\rtimes S_4)$, Magma's OuterFPGroup(AutomorphismGroup(G)). The reason for $C_5$ is that $x$ is only conjugate to $x,x^{-1}$ in $C_{11}\triangleleft G$, but there there are 5 pairs of possible generators like that in $C_{11}$, indistinguishable from each other; the other factor of $Out\ G$ is $Aut(C_2\times C_2\times C_4)$, all of these guys commute with the action.
The representations
The group has 28 orthogonal, 20 symplectic and 8 non-self-dual representations, according to Magma.
The conjugacy classes
There are 1+7+8+5+35=56 conjugacy classes, of elements of order 1,2,4,11,22 respectively. The elements of order 4 are (clearly) not conjugate to their inverses, so these 8 classes account for the 8 non-self-dual representations. We are interested in splitting the other 48 classes into two groups, 28 'orthogonal' and 20 'symplectic'.
The catch
The problem is that the way $Out\ G$ acts on the 35 classes of elements of order 22, it has two orbits according to Magma - one with 30 classes and one with 5. (I think I can see that these numbers must be multiples of 5 without Magma's help, but I don't see the full splitting at the moment; I can insert the Magma code if you guys want it.) Anyway, if I am correct, these 30 classes are indistinguishable from one another, so they must all be either 'orthogonal' or 'symplectic'. So a canonical splitting into 28 and 20 cannot exist.
Edit: However, as Jack Schmidt points out (see comment below), it is possible to predict the number of symplectic representations for this group!