Square roots of elements in a finite group and representation theory

Question

Let $G$ be a finite group. In an an earlier question, Fedor asked whether the square root counting function $r_2:G\rightarrow \mathbb{N}$, which assigns to $g\in G$ the number of elements of $G$ that square to $g$, attains its maximum at the identity element, when $G=S_n$. I gave an affirmative answer using representation theory, which is valid when $S_n$ is replaced by any group that has no symplectic representations. In other words, the argument I gave works whenever any irreducible complex representation of $G$ is either realisable over $\mathbb{R}$ or has non-real valued character. Again in other words (and closest to the actual argument) I need the Frobenius-Schur indicators of all irreducible complex representations to be non-negative.

This is the first of two follow-up questions (the second one being about $n$-th roots), which Pete Clark encouraged me to ask here. I ran a quick computer experiment. There are 1911 groups among the groups of size up to 150 that have a symplectic representation. In 1675 of them, the square root counting function does not attain its maximum at the identity element. Is there a nice (representation theoretic?) criterion that singles out the 300-odd remaining groups? A criterion that includes the previous one as a special case would of course be particularly interesting. in other words, I am asking:

what does it tell you about the group (about its representation theory) if the square root counting function attains its maximum at the identity?

Any brain storming ideas or heuristics are welcome. Criteria that catch some of the remaining groups, even if not all, are also of interest. Thank you in advance!

Answer 1

As pointed out in comments and at this question, this answer is not entirely correct because real/quaternionic does not always give a $\mathbb{Z}/2$ grading. Nonetheless in practice this answer seems to be "mostly right." I'm going to leave it up unmodified because changing it would confuse the comments and other answers too much. Sorry for the mistake!

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If $G$ has a quaternionic representation and $G$ has no complex representations (i.e. reps whose character is complex) then for certain nontrivial central elements the square root function is $\sum_\chi \chi(1)$ which is clearly the maximum possible value and clearly larger than the value on the identity.

The key idea in the proof is the fact that $Z(G)^*$ (the group of characters of the center of $G$) is the universal grading group for the category of $G$ representations.

Let me unpack that a little bit. To every irrep of $G$ you can assign it's "central character", that is an element of $Z(G)^*$. This assignment is multiplicative in the sense if $U$ is a subrep of $V \otimes W$ then the central character of $U$ is the product of the central characters of $V$ and $W$ (this is just because central elements act by scalars). In other words, $\operatorname{Rep}(G)$ is graded by $Z(G)^*$. The claim is that an $H$-grading of $\operatorname{Rep}(G)$ is the same thing as a map $Z(G)^* \to H$.

Why is this true? I'm just going to sketch the proof, in particular I'm just going to talk about the case where $Z(G)$ is trivial, but I'll indicate how the general case works. Since the center is trivial you can find a faithful representation $V$. But then let's look at a really high tensor power of $V$ and ask how it breaks up. We can compute that using character theory. Since the rep is faithful and there's no center, the character of a high tensor power of $V$ is dominated by the value on $1$. Hence any high tensor power of $V$ contains all other irreps. In particular, the $n$-th and $(n+1)$-th powers contain the same irreps and this tells you that the grading group is trivial. In general you want to argue that the contribution of the center dominates the values of inner products (but you need to be a bit careful about non-faithful representations).

The "universal grading group" is used by Gelaki and Nikshych to define the upper central series of an arbitrary fusion category and thereby define nilpotent fusion categories.

Ok, how is this relevant to anything? Well, if you have a group with a quaternionic representation but no complex representations, then the Frobenius-Schur indicator gives a $\pm 1$ grading of $Rep(G)$, namely put the real reps in grade $1$, and the quaternionic ones in grade $-1$. Hence the Frobenius-Schur indicator $s(\chi)$ must be given by the central character $\chi(z)/\chi(1)$ for some central element $z$. Clearly these central elements maximize the square root function.