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I'm trying to find examples of groups $G$ where there is a non-Brauer pairs have an interesting conjugation action on it. I am currently trying the symmetric group in characteristic 2. The center of $kG$, where $k$ has characteristic 2, will be generated by conjugacy class sums.

I believe it's necessary that the elements of the conjugacy class be 2-regular. For $S_3$ and $S_4$, it's sufficient, but those are pretty small cases. Is there anything we can say in general about when a class sum is an idempotent?

Another way to think about it may be that given a conjugacy class $X$,

$\left(\sum\limits_{x\in X} x\right)^2 = \sum\limits_{x\in X}x$ if and only if $\sum\limits_{x \neq y} xy = 0$, which in characteristic 2, is equivalent to an involution on the set $\{ (x,y) | x \neq y \}$.

If you have other examples of groups whose representations have an interesting Brauer pair structure, I'd be interested in those as well.

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For any finite group $G$ to have an idempotent class sum $C^{+}$, where $X^{+}$ denotes the sum of the elements of $X$ in the group algebra) it is certainly necessary that $C$ is a class consisting of elements of odd order. In general in characteristic $p,$ any idempotent in the center of the group algebra is a linear combination of $p$-regular elements as was known to Brauer. I think the statement that for a class sum $X$ we have $X^{2} = X$ (in characteristic $2$ ) if and only if $\sum_{x \neq y} xy =0$ relies on the fact that $x$ is conjugate to $x^{2}$ for all elements $x$ of the underlying conjugacy class. This is true for all elements of odd order in the symmetric group, but need not be true in a general group for elements of odd order.

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