letumsnemesislh

2022-07-08

Intuition behind multiplying (or composing) permutations.
I'm trying to grasp the intuition for permutations and their multiplication. So far this has been my intuitive understanding: A permutation is merely a shuffling of the symbols. Take for example $\sigma ,\pi \in {S}_{4}$ given by, $\sigma =\left(\begin{array}{cccc}1& 2& 3& 4\\ 3& 2& 1& 4\end{array}\right)$ and $\pi =\left(\begin{array}{cccc}1& 2& 3& 4\\ 2& 4& 1& 3\end{array}\right)$
I could rewrite them as a 4-tuple: $\sigma =\left(3,2,1,4\right)$ and $\pi =\left(2,4,1,3\right)$ as permutations of {1,2,3,4} and so $\begin{array}{}\text{(#)}& \pi \circ \sigma =\left(2,4,1,3\right)\circ \left(3,2,1,4\right)=\left(1,4,2,3\right)\end{array}$
I understand how to get the result. I know how to multiply (or compose) two permutations.
My Question: What happened in equation # and what's going on intuitively? What shuffled around when composition happened? What does the result of product mean with respect to $\pi$ and $\sigma$?

Elias Flores

Step 1
We can formulate the composition rule of permutations given in the form
$\begin{array}{r}\pi \circ \sigma =\left(2,4,1,3\right)\circ \left(3,2,1,4\right)=\left(1,4,2,3\right)\end{array}$
as
- Select the items of the left-hand permutation $\pi$ in the order given by the items of the right-hand permutation $\sigma$.
Step 2
In the current example we have the order of the items of the right-hand permutation $\sigma$ as
resulting in $\pi \circ \sigma =\left(1,4,2,3\right)$

Raul Walker

Step 1
I'd say that it happens what the definition says it should happen: if you say that $f=\left({f}_{1},{f}_{2},{f}_{3},{f}_{4}\right)$ is the function such that $f\left(i\right)={f}_{i}$, and if you say that $\pi \circ \sigma$ is the function such that $\left(\pi \circ \sigma \right)\left(i\right)=\pi \left(\sigma \left(i\right)\right)$, then the representation of $\pi \circ \sigma$ will be given by $\left(\pi \circ \sigma {\right)}_{i}={\pi }_{{\sigma }_{i}}$.
Step 2
Of course, you should be careful whe you use this the one-line notation because you are using essentially the same notation as the very common cycle notation.

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