FUNCTION: IdcaOnFree - action of the idCoxeter algebra on words

CALLING SEQUENCE:

IdcaOnFree(e_1, free)

FREE[IdcaOnFree](e_1, free)

PARAMETERS:

e_1 = any element of the idCoxeter algebra

free = any element of the free algebra

SYNOPSIS:

• The IdcaOnFree function realizes the action of the idCoxeter algebra on the free algebra.

• The elementary action of a simple transposition, say (1,2), on a word v is as follows: forget all letters different from 1 and 2, then cross out all successive factors [2,1] of what is left, in any order. The remaining word is of the type v'=w[1$a,2$b]. If a>b then the image of v is the sum of all words obtained from v by replacing its subword v' by w[1$(a-1), 2$(b+1)], ..., w[1$b, 2$a] successively. If a=b, it is sent to 0. If a<b it is sent to minus the image of the word where v' is exchanged into w[1$b, 2$a].

• For example, IdcaOnFree(A[2,1], w[3,1,1,1,2,1,2]) = w[3,1,1,2,2,1,2] + w[3,1,2,2,2,1,2] because v' = [*,1,1,1,*,*,2].

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

EXAMPLES:

> with(FREE):
> IdcaOnFree(A[2,1], w[3,1,1,1,2,1,2]);
 
               w[3, 1, 2, 2, 2, 1, 2] + w[3, 1, 1, 2, 2, 1, 2]