<p>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 $K_n=\{\pi\in S_n \mid \pi \mbox{ is an n-cycle}\}$.</p>

<p>The <a href="http://www.wisdom.weizmann.ac.il/~ranraz/publications/Plogrank.ps" rel="nofollow">paper</a> by Raz and Spieker (see page 4) claimed that it is easy to verify that for any $\pi\in S_n$, $(\sum_{\sigma\in K_n} P_\sigma)P_\pi=P_\pi(\sum_{\sigma\in K_n} P_\sigma)$. </p>

<p>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.) </p>

CC BY-SA 3.0
2011-10-21T13:50:09.350Z
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
CC BY-SA 3.0
2011-10-21T14:52:19.443Z
I think Math.SE might be more appropriate for asking this. :)
Kaveh
CC BY-SA 3.0
2011-10-22T21:25:43.643Z
Tsuyoshi Ito's comment turns out to be exactly what I need. Thank you!
Dan
<algebra>
Commutation between a permutation matrix and the sum of permutation matrices corresponding to n-cycles
Dan
# Commutation between a permutation matrix and the sum of permutation matrices corresponding to n-cycles

(Asked by: username_0 on 2011-10-21)

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](http://www.wisdom.weizmann.ac.il/~ranraz/publications/Plogrank.ps) 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.)

**Tags:** algebra
> **Comments:**
> **username_1**: 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.  
> **username_2**: I think Math.SE might be more appropriate for asking this. :)  
> **username_0**: Tsuyoshi Ito's comment turns out to be exactly what I need. Thank you!  

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## 0 answers

