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Let $G$ be a finite group and $KG$ its group algebra for a field $K$. Let $e$ be an idempotent of $KG$, then the character $\xi_e$ of the module $KG e$ is given by $\xi_e(h)=|C_G(h)| \sum\limits_{g \in C}^{}{a_g}$, where $C_G(h)$ is the centraliser of $h$ in $G$ and $C$ is the conjugacy class of $h^{-1}$ in $G$.

Question: If we now have an irreducible character $\xi$, is there a nice closed formula for the primitive idempotent $e$ such that $\xi $ is the character of $KG e$?

I have only seen a nice formula for central idempotents, which correspond to $V^{dim V}$ for irreducible $V$.

I have not worked with character theory for over 10 years, so sorry if this question is too elementary for MO.

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    $\begingroup$ There is nothing unique about $e$, apart from up to conjugacy in the group algebra. If something is not unique, you should not expect a formula. The interesting thing in mathematics, as opposed to real life, is that it's much easier to find a needle in a haystack if there's only one. On the other hand, in specific situations such as the symmetric group, there are specific formulas, such as those of Young. $\endgroup$
    – Dave Benson
    Commented Apr 13, 2023 at 20:48
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    $\begingroup$ Over the complex numbers there is a Fourier inversion theorem that you can apply to a matrix unit E_11 from a particular irreducible representation to get a formula for a primitive idempotent. Im not sure how useful this formula is. I guess it generalizes in a straightforward way to a splitting field in nonmodular characteristic. $\endgroup$
    – Benjamin Steinberg
    Commented Apr 13, 2023 at 23:25
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    $\begingroup$ The Fourier inversion theorem though gives you an answer in terms of the matrix coefficients of the unitary irreducible representations not in terms of the character. $\endgroup$
    – Benjamin Steinberg
    Commented Apr 13, 2023 at 23:32

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I am not sure this is worth making an answer, rather than a comment, but in the complex case, we may write the primitive idempotent $e$ in the form $e = \frac{e_{\chi}}{\chi(1)} + \sum_{g \in G} \alpha_{g}g,$ where $\chi$ is the unique complex irreducible character with $\chi(e) \neq 0$, $\sum_{g \in C} \alpha_{g} =0$ for each conjugacy class $C$ of $G$, and $e_{\chi}$ is the centrally primitive idempotent associated to $\chi$.

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