GAP Manual: 63.22. DecomposedLeftCells

DecomposedLeftCells( W, c, mue )

given a list c of reduced words in the Weyl group W with corresponding matrix mue of highest coefficients of the corresponding Kazhdan-Lusztig polynomials, it returns a list of pairs. The first component of each pair consists of the reduced words in the list c 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> WeylElements(W);;
    #I  Order = 12
    gap> c := Iterated( last, Concatenation );
    [ [  ], [ 1 ], [ 2 ], [ 1, 2 ], [ 2, 1 ], [ 1, 2, 1 ], [ 2, 1, 2 ],
      [ 1, 2, 1, 2 ], [ 2, 1, 2, 1 ], [ 1, 2, 1, 2, 1 ],
      [ 2, 1, 2, 1, 2 ], [ 1, 2, 1, 2, 1, 2 ] ]
    gap> mue := WeylMueMat( W, c );
    #I  Order = 12
    [ [ 0 ], [ 1, 0 ], [ 1, 0, 0 ], [ 0, 1, 1, 0 ], [ 0, 1, 1, 0, 0 ],
      [ 0, 0, 0, 1, 1, 0 ], [ 0, 0, 0, 1, 1, 0, 0 ],
      [ 0, 0, 0, 0, 0, 1, 1, 0 ], [ 0, 0, 0, 0, 0, 1, 1, 0, 0 ],
      [ 0, 0, 0, 0, 0, 0, 0, 1, 1, 0 ],
      [ 0, 0, 0, 0, 0, 0, 0, 1, 1, 0, 0 ],
      [ 0, 0, 0, 0, 0, 0, 0, 0, 0, 1, 1, 0 ] ]
    gap> DecomposedLeftCells( W, c, mue );
    [ [ [ [  ] ], [ [ 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
                 ] ] ],
      [ [ [ 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
                 ] ] ], [ [ [ 1, 2, 1, 2, 1, 2 ] ], [ [ 0 ] ] ] ]

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