# Math Genius: Primitive idempotents in symmetric group rings

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
$$sigmasigma_1sigma=sigma_1notin Rsigma,$$
$$Rightarrow:$$ 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.