FUNCTION: NilFree2Free - plaxification map

CALLING SEQUENCE:

NilFree2Free(free)

FREE[NilFree2Free](free)

PARAMETERS:

free = any element of the free algebra

SYNOPSIS:

• The NilFree2Free function transforms each word seen as a reduced decomposition into a word, according to the plaxification map. This map is defined recursively by: plaxify(w[i,j,...]) = w[i, SgaOnFree(s_i, plaxify(w[j,...]))] where s_i is the simple transposition (i,i+1).

• The plaxification map has the property to send a nilplactic class (of reduced decompositions) onto a plactic class; it moreover preserves the Q-symbol of Schensted.

• Whenever there is a conflict between the function name NilFree2Free and another name used in the same session, use the long form FREE['NilFree2Free'].

EXAMPLES:

> with(FREE):
> NilFree2Free(w[2,1,2,1]);
 
                                w[2, 1, 3, 1]
 
> NilFree2Free(w[2,3,1] + w[2,1,3]);
 
                           w[2, 1, 2] + w[2, 2, 1]
 
> r := map(z -> w[op(z)], Perm2ListRd([2,4,3,1]));
 
           r := [w[1, 2, 3, 2], w[3, 1, 2, 3], w[1, 3, 2, 3]]
 
> r := map(NilFree2Free, r);
 
           r := [w[1, 1, 3, 1], w[3, 1, 1, 1], w[1, 3, 1, 1]]
 
> map(Word2Tab, r);
 
            [[[3], [1, 1, 1]], [[3], [1, 1, 1]], [[3], [1, 1, 1]]]