I was watching a video on tranpositions and it isn't obvious to me why when decomposing a cycle, we swap the position of the elements in the cycle instead of swapping the elements themselves. I would like if someone could explain to me what is really going on when we do these transpositions.
So for instance, to write $(4, 1, 2, 3)$ as a disjoint transpositions, one could do (the textbook way):
$$(4, 3)(4, 2)(4, 1)$$
while we could also do $$(3, 4)(2,3)(1,2)$$
as in the video. Note that I flipped the order because I calculate transpositions from right to left. So my convention is I evaluate $(1,2)$, then $(2,3)$, and finally $(3,4)$.
In the video I linked (I already timed it exactly, so you won't have to skip), he drew this matrix thing I have never seen before in any textbook.
$$\begin{pmatrix} 1 &2 &3 &4 \\ 1&2 &4 &3 \\ 1 &4 &2 &3 \\ 4&1 &2 &3 \end{pmatrix}$$
To my understanding, he swapped $4$ and $3$ initially, and then swapped $4,2$ in the middle, but he writes $(2, 3)$ indicating he is only swapping the positions, not permutation the elements. Can someone formally write out what is going on here?
For instance, I tried formalizing and conjecturing that he is multiplying $(1, 2,3,4)(3,4)$ and then keep decomposing, and obviously I was wrong.
In my eyes the matrix represents exactly the same operations that you wrote down in cycle notation: the 2nd line of the matrix indicates that the two numbers at the positions 3 and 4 are swapped - (3,4) in cycle notation. Next in the 3rd line the two numbers at the position 2 and 3 are swapped - (2,3). In last line the numbers at the positions 1 and are swapped - (1,2). Altogether $(1,2)(2,3)(3,4)$, which is correct.
Remark: you are talking about disjoint decomposition into transpositions. But all the decompositions appearing are not disjoint. In fact a cycle can only be decomposed into disjoint transpositons if it is already a transposition.