Commutation between a permutation matrix and the sum of permutation matrices corresponding to n-cycles

Let $S_n$ be the set of all permutations of $n$ elements. Consider the regular representation of $S_n$ in $GL(\mathbb{R}^{n!})$ by $S_n\ni \pi \rightarrow P_\pi$: $(P_\pi)_{\sigma\tau}=1$ if $\pi \circ \tau=\sigma$ and 0 otherwise. Let $K_n=\{\pi\in S_n \mid \pi \mbox{ is an n-cycle}\}$.

The paper by Raz and Spieker (see page 4) claimed that it is easy to verify that for any $\pi\in S_n$, $(\sum_{\sigma\in K_n} P_\sigma)P_\pi=P_\pi(\sum_{\sigma\in K_n} P_\sigma)$.

I would like to request for a pointer to the relevant facts that can be used to prove this fact. (Please note that I don't have much background on this -- I'm pretty weak at Algebra and know almost nothing about representation theory. So, any relevant facts that can be found in basic textbooks will be much appreciated.)

• Does it help if the equation to prove is rewritten to $P_{\pi^{-1}}(\sum_{\sigma\in K_n} P_\sigma)P_\pi=\sum_{\sigma\in K_n} P_\sigma$? The simple fact needed to prove this equation is that a conjugate of an n-cycle is an n-cycle. – Tsuyoshi Ito Oct 21 '11 at 13:50
• I think Math.SE might be more appropriate for asking this. :) – Kaveh Oct 21 '11 at 14:52
• Tsuyoshi Ito's comment turns out to be exactly what I need. Thank you! – Danu Oct 22 '11 at 21:25