FUNCTION: PermOnTab - action of a permutation on a product of rows

CALLING SEQUENCE:

PermOnTab(perm, tab)

TAB[PermOnTab](perm, tab)

PARAMETERS:

perm = any list denoting a permutation

tab = any list of lists denoting a product of rows

SYNOPSIS:

• The PermOnTab function plactically permutes rows in a product of rows, say tab (not necessarily a tableau, but the image of a tableau under a row-permutation).

• A simple plactic row-transposition transforms a two-row contre-tableau into the two-row tableau which is obtained from it by Schensted algorithm, or conversely. This transformation is also given by the jeu de taquin.

• For example 1 3 4 6 and 3 6 are interchanged. 2 5 1 2 4 5

• When used with the extra argument 'nil', it nilplactically permutes rows, exchanging for instance [[2], [1,2]] and [[1,2], [1]], while a plactic row-transposition exchanges [[2], [1,2]] with [[2,2], [1]]. Whenever the tableau is not a reduced decomposition, it returns [].

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

EXAMPLES:

> with(TAB):
> PermOnTab([2,1], [[1,3,4,6], [2,5]]);
 
                            [[3, 6], [1, 2, 4, 5]]
 
> PermOnTab([2,1], [[4], [3,4]], 'nil');
 
                                [[3, 4], [3]]