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Let ${\rm Irr}(G)$ be the set of complex irreducible characters of a finite group $G$. The Frobenius-Schur indicator of $\chi\in{\rm Irr}(G)$ is defined to be $\epsilon(\chi):=\frac{1}{|G|}\sum_{g\in G}\chi(g^2)$. It is known that $\epsilon(\chi)=-1,0,1$ and that $\epsilon(\chi)\ne0$ if and only if $\chi$ is real valued. Moreover $$ \sum\limits_{\chi\in{\rm Irr}(G)}\hspace{-.1cm}\epsilon(\chi)\chi(g)=|\{x\in G\mid x^2=g\}|,\quad\mbox{for all $g\in G$.} $$

Consider the following class functions on $G$: $$ a(g):=\sum_{\chi\in{\rm Irr}(G)}\frac{\chi(g)}{\chi(1)},\quad b(g):=\sum_{\chi\in{\rm Irr}(G)}\frac{\epsilon(\chi)\chi(g)}{\chi(1)},\quad c(g):=\sum_{\chi\in{\rm Irr}(G)}\frac{\epsilon(\chi)^2\chi(g)}{\chi(1)}. $$ For obvious reasons the average value of each of $a,b,c$ is $1$. Moreover it is known that $$ \begin{aligned} |G|a(g)&=|\{x,y\in G\mid x^{-1}y^{-1}xy=g\}|\quad\mbox{and}\\ |G|c(g)&=|\{x,y\in G\mid xy^{-1}xy=g\}|. \end{aligned} $$ So $|G|a$ and $|G|c$ take non-negative integer values. In the few cases I checked, so too does $|G|b$.

Question: does $|G|b$ count anything? More specifically, is there a 2-variable word $w$ such that $|G|b(g)=|\{x,y\in G\mid w(x,y)=g\}|$ for all $g\in G$?

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    $\begingroup$ I don't think it helps at all, but I think it is also the case that $c(g) \neq 0$ if and only if $g$ is expressible as a product of two squares in $G$. $\endgroup$
    – Geoff Robinson
    Jul 5, 2018 at 21:01
  • $\begingroup$ Hi John, do you have a reference for the formulas for $|G|a(g)$ and $|G| c(g)$? $\endgroup$
    – James
    Mar 18, 2020 at 12:31
  • $\begingroup$ Consider $\#\{x,y\in G\mid x^{-1}y^{-1}xy=g\}$. This is the coefficient of $g$ in $Z:=\sum K^-K^+$, where $K$ ranges over the conjugacy classes of $G$, $K^-=\sum_{k\in K}k^{-1}$ and $K^+=\sum_{k\in K}k$ in the complex group algebra ${\mathbb C}G$. $\endgroup$
    – John Murray
    Mar 25, 2020 at 10:56
  • $\begingroup$ Now $Z$ is central in ${\mathbb C}G$. So $Z=\sum_{\chi\in{\rm Irr}(G)}\omega_\chi(Z)e_\chi$, where $\omega_\chi(Z):=\chi(Z)/\chi(1)$ is the central character of $\chi$ and $e_\chi$ is the central idempotent $(\chi(1)/|G|)\sum_{h\in G}\chi(h^{-1})h$. The key point is that $\omega_\chi$ is an algebra homomorphism from the centre of ${\mathbb C}G$ to ${\mathbb C}$. So $\omega_\chi$ is additive and multiplicative. Expanding out $\omega_\chi(\sum K^-K^+)$, and using the second orthogonality relation, we get the expression for $|G|a(g)$. Similarly for $|G|c(g)$. $\endgroup$
    – John Murray
    Mar 25, 2020 at 10:57
  • $\begingroup$ Thanks @JohnMurray. $\endgroup$
    – James
    Jun 6, 2020 at 11:43

1 Answer 1

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The group $๐บ=\mathtt{PerfectGroup}(43008,10)โ‰…๐ถ_2^7\rtimes ๐‘†๐ฟ(2,7)$ has an element $g$ such that $๐‘(๐‘”)<0$. Hence, $b$ cannot count something. The first such example was found by Gabriel Navarro.

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