Dears,

Sorry for inconvenience,

There is an error in the explaining of the requested
function. In the definition of $w_i$'s, it must be as
follows:

$w_i=[x_{j_1}^{a_1},\dots,x_{j_{t_i}}^{a_{t_i}}]$
  where $a_q$'s are in $\{ -1, 1 \}$.

Now the question is as follows:

Let $x_1,\dots, x_n$ be the generators of the free
group of rank $n$ and $k$ be a positive integer.

I look for a function with the following input and
output in GAP (or somthing like this):

the input of the function is : $x_1,\dots, x_n$ and
$k$.

the output is the list $w_1,\dots, w_s$ satisfying
the
following conditions:

 $(x_1 \cdots x_n)^k=x_1^k \cdots x_n^k w_1 \cdots 
w_s$  for 

all $i=1,\dots, s$,  

$w_i=[x_{j_1}^{a_1},\dots,x_{j_{t_i}}^{a_{t_i}}]$

where

$a_q$'s are in $\{ -1, 1 \}$

$j_1, \dots, j_{t_i}$ in $\{ 1, \dots, n  \}$ and

$t_1 \leq t_2 \leq \dots \leq t_s$. ( "$[y_1,\dots,
y_m]$" donotes the usual left normed commutator 
defined inductively
by  $[x,_0 y]=x$, $[x,_1 y]:=[x,y]:=x^{-1}y^{-1}xy$
and $[y_1,\dots,y_m]=[[y_1,\dots,y_{m-1}],y_m]$ for
all $m>1$.

Of course the commutators $w_i$'s are depend to the
way of collecting $x_i$'s and in practice they are
not necessarily unique.

Thanks for any help.
With best wishes
Alireza Abdollahi

=====
Alireza Abdollahi
Department of Mathematics
University of Isfahan,
Isfahan 81744,Iran
e-mail: alireza_abdollahi@yahoo.com
URL: http://www.abdollahi.8m.net

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