Composition of permutation operators Im reading a course on second quantization and they say that the composition of permutation operators is:
$$ P_{\sigma} \circ P_{\sigma'} = P_{\sigma' \circ \sigma} $$
But for me it should be:
$$ P_{\sigma} \circ P_{\sigma'} = P_{\sigma \circ \sigma '} $$
I take an example.
Imagine I work with 3 particles.
I have $| \psi \rangle = |1:u_1,2:u_2,3:u_3 \rangle $
I apply : $P_{312} \circ P_{132} $ on it.
$$ P_{132} |1:u_1,2:u_2,3:u_3 \rangle = |1:u_1,3:u_2,2:u_3 \rangle=|1:u_1,2:u_3,3:u_2 \rangle$$
Thus:
$$ P_{312} \circ P_{132} |1:u_1,2:u_2,3:u_3 \rangle  = P_{312} |1:u_1,2:u_3,3:u_2 \rangle = |1:u_3,2:u_2,3:u_1 \rangle$$
I remark that : $P_{321} \circ P_{132} = P_{321}$
And I have the permutations $321 \circ 132 = 321$
So why do we have to "invert" the composition in the first equations written ? I don't get it.
I don't think it is a mistake from the course I read because they wrote to be careful with the inversion of permutations.
 A: This is a perennial problem in the literature of permutation groups and their action on sets. There are two ways in which permutations can act on a ket $|a_1,a_2,a_3>$. One can interpret $S_1=(12)$ as the instruction to interchange the label in place 1 with that in place 2, or as the instruction to interchange the label with subscript 1 with that of subscript 2. For the first move it makes no difference: in each case  $S:|a_1,a_2a_3> \to |a_2,a_1,a_3>$. It makes a difference at the next move. Does $S_2=(13)$ take $|a_2,a_1,a_3> \to |a_3,a_1,a_2>$ or to $|a_2,a_3,a_1>$?  If $S_1\circ S_2=S_3$ in the first interpretation, then $S_2\circ S_1=S_3$ in the second.  It's like the way body-fixed and space-fixed rotation obey opposite composition rules in mechanics.
Beware  also that, for historical reasons, some books read the composion of permutions from left to right -- in $S_1S_2$ they first act by $S_1$ and then by $S_2$. I wrote "$\circ$'' to imply composition in which $S_1\circ S_2$ means first $S_2$ then $S_1$. 
A: Using $P[321]=P[13]$ and $P[132]=P[23]$ we have
\begin{align}
P[23]_R(123)&=(132)\, ,\\
P[13]_R(132)&=(231)\, ,\\
P[132]_R(123)&=(231)
\end{align}
so $P[13]_R\circ P[23]_R=P[132]_R$
since $P[13]_R$ etc change the slots, not the symbols, and $P[abc]_R$ means slot $a \to$ slot b, slot $b\to $ slot c, slot $c\to $ slot a. 
If you want to act on the symbols rather than the slots:
\begin{align}
P[23]_L(123)&=(132)\, ,\\
P[13]_L(132)&=(312)\, ,\\
P[132]_L(123)&=(312)
\end{align}
you get $P[13]_L\circ P[23]_L=P[132]_L$.  Hence you see that 
$P[abc]_L=P^{-1}[abc]_R$.
The first action is an example of a right action, and the second is an example of a left action. 
In the physics parlance, one is an active and the other is a passive transformation. This is similar to $x\to x+a\Rightarrow \psi(x)\to \psi(x-a)$.
