FUNCTION: SgaOnFree - action of the symmetric group algebra on words

CALLING SEQUENCE:

SgaOnFree(e_1, free)

FREE[SgaOnFree](e_1, free)

PARAMETERS:

e_1 = any element of the symmetric group algebra

free = any element of the free algebra

SYNOPSIS:

• The SgaOnFree function realizes the action of the symmetric group algebra on the free algebra. This action is compatible with plactic congruences and Schensted algorithm.

• The elementary action of a simple transposition, say A[2,1], 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]. Then the image of v is obtained by replacing its subword v' by w[1$b, 2$a].

• In particular, the image of a tableau (resp. contretableau) is a tableau (resp. contretableau).

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

EXAMPLES:

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