Date: Sep 8, 2013 12:03 PM
Author: fom
Subject: Re: abelian square

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/computer-science/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.