Number of n-th roots of elements in a finite group and higher Frobenius-Schur indicators

Question

This is the second follow-up to this question on square roots of elements in symmetric groups and is concerned with generalisations to $n$-th roots. Let $G$ be a finite group and let $r_n(g)$ be the number of elements $h\in G$ such that $h^n = g$. In other words, $$r_n(g) = \sum_{h\in G}\delta_{h^n,g},$$ where $\delta$ is the usual Kronecker delta. In a comment to my answer to the above mentioned question, Richard Stanley notes that if $G=S_m$, then $r_n(g)$ attains its maximum at the identity element of $G$. My question is: how far does this generalise and what exactly does it tell us about $G$? This should be primarily a question about higher Frobenius-Schur indicators. Let me elaborate a bit.

The function $r_n$ is clearly a class function on $G$ and, upon taking its inner product with all irreducible characters of $G$, one finds that $$r_n(g) = \sum_\chi s_n(\chi)\chi(g),$$ where the sum runs over all irreducible complex characters of $G$ and $s_n(\chi)$ is the $n$-th Frobenius-Schur indicator of $\chi$, defined as $$s_n(\chi) = \frac{1}{|G|}\sum_{h\in G}\chi(h^n).$$ When $n=2$, the Frobenius-Schur indicator is equal to 0,1 or -1 and carries explicit information about the field of definition of the representation associated with $\chi$.

What do higher Frobenius-Schur indicators tell us about the representations and, by extension, about the group? What do we know about their values? Have higher Frobenius-Schur indicators been studied in any detail?

For additional focus:

Given $n\in \mathbb{N}$, for what groups $G$ do we have $\max_g \; r_n(g) = r_n(1)$? For what groups does this hold for all $n$?

As noted by Richard Stanley, the latter is true for all symmetric groups. It is also easy to see that the set of groups with this property is closed under direct products, and that all finite abelian groups possess this property.

Answer 1

Here are some things you probably know. For a representation $W$ of $G$, let $\text{Inv}(W)$ denote the subspace of $G$-invariants. For an irreducible representation $V$ with character $\chi$, the F-S indicator $s_2(\chi)$ naturally appears in the formulas

$$\dim \text{Inv}(S^2(V)) = \frac{1}{|G|} \sum_{g \in G} \frac{\chi(g)^2 + \chi(g^2)}{2}$$

and

$$\dim \text{Inv}(\Lambda^2(V)) = \frac{1}{|G|} \sum_{g \in G} \frac{\chi(g)^2 - \chi(g^2)}{2}.$$

More precisely the F-S indicator is their difference, while their sum is $1$ if $V$ is self-dual and $0$ otherwise. The corresponding formulas involving $s_3(\chi)$ are

$$\dim \text{Inv}(S^3(V)) = \frac{1}{|G|} \sum_{g \in G} \frac{\chi(g)^3 + 3 \chi(g^2) \chi(g) + 2 \chi(g^3)}{6}$$

and

$$\dim \text{Inv}(\Lambda^3(V)) = \frac{1}{|G|} \sum_{g \in G} \frac{\chi(g)^3 - 3 \chi(g^2) \chi(g) + 2 \chi(g^3)}{6}.$$

Here the F-S indicator $s_3(\chi)$ naturally appears in the sum, not the difference, of these two dimensions. $T^3(V)$ decomposes into three pieces, and the third piece is (Edit, 9/26/20: two copies of) the Schur functor $S^{(2,1)}(V)$, which therefore satisfies

$$\dim \text{Inv}(S^{(2,1)}(V)) = \frac{1}{|G|} \sum_{g \in G} \frac{ \chi(g)^3 - \chi(g^3)}{3}.$$

So $s_3(\chi)$ constrains the dimensions of these spaces in some more mysterious way than $s_2(\chi)$ does. The sum

$$\dim \text{Inv}(T^3(V)) = \frac{1}{|G|} \sum_{g \in G} \chi(g)^3$$

tell us whether $V$ admits a "self-triality," and this dimension is an upper bound on $s_3(\chi)$. If $V$ is self-dual, this is equivalent to asking whether there is an equivariant bilinear map $V \times V \to V$, which might be of interest to somebody. If this dimension is nonzero then $s_3(\chi)$ gives us information about how a triality behaves under permutation.

The situation for higher values of $3$ is worse in the sense that the bulk of the corresponding formulas are not completely in terms of F-S indicators but in terms of inner products of F-S indicators and their interpretation will only get more confusing. Already I don't know of many applications of triality (in fact I know exactly one: http://math.ucr.edu/home/baez/octonions/node7.html).

Answer 2

If $n > 2$, there is NO absolute upper bound on the "higher" F.S. indicator $s_n(\chi)$. This is Problem 4.9 in my character theory book. (A hint is given there.)

Answer 3

The concept of higher Frobenius-Schur indicators has been generalized to far broader contexts than just group representations, and these have proven extremely useful.

Linchenko and Montgomery generalized the second indicators to semisimple Hopf algebras, and Kashina, Sommerhauser, and Zhu generalized higher indicators to the same. They were then defined for spherical fusion categories by Ng and Schauenburg, and have been used in problems ranging from classifications of low dimensional (quasi-)Hopf algebras; relating exponents and dimensions; obtaining new gauge invariants; and much more— the intro for this paper by Negron and Ng gives a pretty solid rundown on these generalizations and their uses. Some of the generalizations do not require semisimplicity, which for the most part means the invariants are only of the regular representation (rather than arbitrary fin. dim. reps). Searching for "frobenius-schur indicators" on the math arxiv should pull up quite a number of results.

When applied to the quasi-triangular Hopf algebras D(G), the Drinfeld double of the group G, you get a number of new group invariants which are, loosely speaking, based on how many roots (of a given element) are sent to roots (of the same element) under the action of each element under the regular representation. It then becomes quite interesting to investigate how this depends on the given element. Sometimes it only depends on the cyclic subgroup it generates (such as for all regular p-groups), sometimes it doesn't (such as for some irregular p-groups and the Monster group). Ultimately these can be tied into how the group is pieced together from its one-element centralizers, which if you know much about the representation theory of D(G) is quite sensible for an invariant of Rep(D(G)): representations are given by inducing representations from centralizers to the whole group.

Quite recently, Barter, Jones, and Tucker showed how these indicators effectively govern the structure of certain annular Temperley-Lieb-Jones modules in the fusion categories. Such objects are specified by rotation eigenvalues (which all have a predictable modulus, provided you know the twists), and the indicators govern the coefficients of the polynomials that dictate the multiplicities of these eigenvalues.