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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.

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Thanks, Denis, for the edit! Well spotted. –  Alex B. Oct 18 '10 at 16:02
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2 Answers

up vote 4 down vote accepted

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. Of course $T^3(V)$ decomposes into three pieces, and the third piece $S^{(2,1)}(V)$ satisfies

$$\dim \text{Inv}(S^{(2,1)}(V)) = \frac{1}{|G|} \sum_{g \in G} \frac{4 \chi(g)^3 - 4 \chi(g^3)}{6}.$$

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

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

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).

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Dear Qiaochu, thank you! I didn't know about triality, very interesting! I will have to think about this a bit. I like "higher values of 3". –  Alex B. Feb 9 '11 at 3:53
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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.)

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Dear Marty, thank you. I like that problem very much, and I also set it to my students when I taught representation theory last year. –  Alex B. Feb 2 '12 at 23:08
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