GAP Manual: 63.23. LeftCells

LeftCells( W )

returns a list of pairs. The first component of each pair consists of the reduced words in the Weyl group W which lie in one left cell C, the second component consists of the corresponding matrix of highest coefficients mu_(y,w), where y,w are in C.

    gap> W := Weyl( CartanMat("G",2) );;
    gap> LeftCells(W);
    #I  Order = 12
    #I  R(w) = [  ] :  Size = 1,  1 new cell
    #I  R(w) = [ 1 ] :  Size = 5,  1 new cell
    #I  R(w) = [ 1, 2 ] :  Size = 1,  1 new cell
    #I  R(w) = [ 2 ] :  Size = 5,  1 new cell
    [ [ [ [  ] ], [ [ 0 ] ] ],
      [ [ [ 1 ], [ 2, 1 ], [ 1, 2, 1 ], [ 2, 1, 2, 1 ], [ 1, 2, 1, 2, 1 ]
             ],
          [ [ 0 ], [ 1, 0 ], [ 0, 1, 0 ], [ 0, 0, 1, 0 ], [ 0, 0, 0, 1, 0
                 ] ] ], [ [ [ 1, 2, 1, 2, 1, 2 ] ], [ [ 0 ] ] ],
      [ [ [ 2 ], [ 1, 2 ], [ 2, 1, 2 ], [ 1, 2, 1, 2 ], [ 2, 1, 2, 1, 2 ]
             ],
          [ [ 0 ], [ 1, 0 ], [ 0, 1, 0 ], [ 0, 0, 1, 0 ], [ 0, 0, 0, 1, 0
                 ] ] ] ]

This function requires the package "weyl" (see RequirePackage).