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Let $G$ be a finite group of exponent $n$ and let $d\mid n$. Consider the class function $$ f(g) = \begin{cases} 1 & g^d =1\\0&\textrm{otherwise}. \end{cases} $$ As a class function $f$ can be written as $f= \sum_{\chi} c_{\chi} \chi$, with $$ c_{\chi} =\left<f,\chi \right> = \frac{1}{|G|} \sum_{g^d=1} \chi(g) $$ and where $\chi$ runs over the irreducible characters of $G$.

Q. Is there an explicit formula for $c_{\chi}$?

The question is motivated by the fact that when $G$ is cyclic there is a nice formula: Write $n=dk$, then $f = \sum_{\chi^k=1} \chi$.

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    $\begingroup$ Haven't you given an explicit formula for $c_\chi$? $\endgroup$
    – LSpice
    Commented Dec 12, 2017 at 22:26
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    $\begingroup$ When d=2 this is related to the Frobenius-Schur indicator. $\endgroup$
    – Benjamin Steinberg
    Commented Dec 12, 2017 at 22:36
  • $\begingroup$ @LSpice I don’t think so since I can’t evaluate it in general. $\endgroup$
    – Lior Bary-Soroker
    Commented Dec 12, 2017 at 22:43
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    $\begingroup$ Isn't this just the Fourier transform? $\endgroup$
    – Igor Rivin
    Commented Dec 13, 2017 at 3:04
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    $\begingroup$ Your $f$ can be written in terms of Adams operations as $f=n^{-1}\psi^d(\chi_1)$, so you might find it helpful to consider the relationship between Adams operations and exterior powers. $\endgroup$
    – Neil Strickland
    Commented Dec 13, 2017 at 9:50

1 Answer 1

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This is a partial answer. Frobenius investigated such Fourier coefficients to prove that the number of solutions to $x^d=1$ is divides the order of the group. He showed your class function is a virtual character, if I understood correctly, so all your Fourier coefficients are integers. The paper of Frobenius is in German so I can't read it. The paper http://u.cs.biu.ac.il/~vishne/publications/S0219498811004690.pdf, and its references, should give you some of what you want.

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