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12/4/12


Re: abelian square
Posted:
Sep 8, 2013 12:03 PM


On 9/8/2013 7:16 AM, Dieter von Holten wrote: > hi folks, > > i found 'abelian squares' in the context of patterns of n symbols of length 2n like 'abcd.cdba' where 'cdba' is a permutation of the first half 'abcd'. > however, in those texts it is not explained (maybe just not clear enough for me) what is 'abelian' and where is the 'square' ?? > > any hints ? > > thanks > dvh >
Looking at the definition in
http://arxiv.org/pdf/0807.5028v1.pdf
both
abcd.cdab
cdab.abcd
satisfy the constraint that the second string is a permutation of the first.
Since this is true of all finite strings, one would expect that "abelian" refers simply to the fact that the two strings commute over the separator, '.', relative to the definition.
As for being a "square", that would seem to be a trickier guess. If you look at section 2 of the paper,
http://poncelet.math.nthu.edu.tw/disk5/js/computerscience/on_abelian_squares_and_substitut.pdf
you will find that the Parikh vector mentioned in the first paper is the basis for an equivalence class as defined in the second paper.
Relative to this equivalence class, one has
[x] = [x']
[x][x'] = [x]^2 = [x']^2
So, this would be my guess concerning the sense of "square" in the name.



