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?