Math Genius: Primitive idempotents in symmetric group rings

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I am reading on a paper about the character theory of symmetric groups $S_n$. There is a claim:

Set $sigma$ be an idempotent in the ring $R(S_n)$. Then $sigma$ is primitive iff for any $rin R(S_n)$,
$$sigma r sigma=xi_r^sigma sigma$$
where $xi_r^sigma$ is real.

$Leftarrow:$ If $sigma$ is not primitive, then we have the orthogonal-idempotent decomposition $sigma=sigma_1+sigma_2$. Note that
$$sigmasigma_1sigma=sigma_1notin Rsigma,$$
which contradicts.

$Rightarrow:$ Here is the part I meet with problems. I cannot find a way to draw a contradiction, or to use the property of ‘primitive’ to show the right side.

In addition, I wonder the claim is true or not on the other group. Or to say, do we need some specific proposition of symmetric group to prove?

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