Adjacent Transpositions of Permutations.

[Ad123q]

Hi,

Was wondering if anyone could explain to me what an adjacent transposition is (in relation to permutations, cycles etc).

I know what a transposition is, eg the product of transpositions for (34785) would be (35)(38)(37)(34).
I don't know what an adjacent transposition is though.

Cheers in advance.

 

[HallsofIvy]

An "adjacent transoposition" is just the transposition of two adjacent elements. (23) is an "adjacent transposition" because 2 and 3 are right next to each other (adjacent).

 

[Ad123q]

Thanks, so if I was to express (34785) as the product of adjacent transpositions, would this just be (34) ?

 

[Ad123q]

anyone ?

 

[HallsofIvy]

Thanks, so if I was to express (34785) as the product of adjacent transpositions, would this just be (34) ?

No, of course, not. Why would you think it would reduce to just that? (34785) means "3 changes to 4, 4 changes to 7, 7 changes to 8, 8 changes to 5, and 5 changes back to 3". It is a shorthand for the permutation (12345678)->(12473685). Probably the simplest is just to work from left to right: 1 and 2 are fixed so swap 3 and 4 to get (12435678). Now I need to work that "5" back to the last position so use
1) (56) to get (12436578)
2) (57) to get (12436758)
3) (58) to get (12436785)

Now (67) is obvious. It gives (12437685). And finally, (37) gives (12473685), the final result. That is, (34785) is given by (34)(56)(57)(58)(67)(37).

 

[cup]

Writing a cycle as a product of permutations is easy:

(34785) = (34)(47)(78)(85)

But (47) and (85)( =(58) ) are not adjacent, so we rewrite them as:

(47) = [(56)(45)]^-1(67)[(56)(45)] = [(45)(56)](67)[(56)(45)]
(58) = [(67)(56)]^-1(78)[(67)(56)] = [(56)(67)](78)[(67)(56)]

(Make a drawing to see why this works)

So:

(34785) = (34)(45)(56)(67)(56)(45)(78)(56)(67)(78)(67)(56)

Note: You "follow the permutation" from right to left, in the notation that I use.

 

[Ad123q]

ah, thanks.

Apologies for my ignorance ;)