Write a transposition as a product of adjacent transpositions

Question

I just read that any transposition can be written as a product of adjacent transpositions. I thought that I knew the right proof of this, but then I read that $\tau_{i,j} = \tau_{i,i-1} \circ...\circ \tau_{j+1,j},$ where $i > j$ and $\tau_{a,b}$ permutes the a-th and b-th element.

Somehow I don't think that this is true or is it?

is: Write a permutation as a product of adjacent transpositions? — janmarqz, Jan 07 '15 at 20:13
ok, with an example: is it something like: $(123)=(13)(12)$? — janmarqz, Jan 07 '15 at 20:15
This is not true. For instance, $(42) \neq (43)(32)$. Plug in $2$ (on either side) and you do get $4$, but if you plug in $4$ on the left hand side you get $3$. — breeden, Jan 07 '15 at 20:17
this question asks if: $(ij) = (i \quad i-1)...(j+1 \quad j)$ holds? — Florence Maluda, Jan 07 '15 at 20:17

score 5 · Accepted Answer · answered Jan 07 '15 at 20:24

5

The way I go about solving something like this is by drawing a bunch of $x$'s to denote generic numbers and $o$'s to denote the numbers I want to transpose in a line. Then the game is to only be allowed to do adjacent transpositions if to achieve your desired affect. Your first goal is to get your first $o$ to where the second $o$ is, and you can do that by what you have written. After playing around like this you can convince yourself that the formula should be:

$$ (i\, j) = (j+1\,\,\,j)\dots(i-1\,\,\,i-2)(i\,\,\,i-1)(i-1\,\,\,i-2)\dots(j+1\,\,\,j).$$

Now you just have to prove this formula is correct.

answered Jan 07 '15 at 20:24

breeden

1,540

1

with this recipe, how do you factorize $(1,5)$? – janmarqz Jan 07 '15 at 23:56
1

@janmarqz Good question, I should have provided an example to clarify. $(1,5) = (2,1)(3,2)(4,3)(5,4)(4,3)(3,2)(2,1)$. Check it out, it works! – breeden Jan 08 '15 at 01:06
please write it for $(1,3)$ – janmarqz Jan 08 '15 at 01:08
:) $(1,3) = (2,1)(3,2)(2,1)$. – breeden Jan 08 '15 at 01:09
Hehehhe, I recognized! It is the conjugation class! Thanks. – janmarqz Jan 08 '15 at 01:14
janmarqz, good observation! I didnt recognize that, but this is indeed nothing more than conjugating a single transposition. – breeden Jan 09 '15 at 04:21
BTW, this is similar to what https://en.wikipedia.org/wiki/Cyclic_permutation#Properties says. – An5Drama Feb 16 '24 at 00:58

Write a transposition as a product of adjacent transpositions

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