83.6 LeftCellRepresentation

LeftCellRepresentation( W , cell )

returns a list of matrices giving the left cell representation of the Iwahori-Hecke algebra W. The argument cell is a pair with first component a list of reduced words which form a left cell, and second component the corresponding matrix of highest coefficients of the corresponding Kazhdan-Lusztig polynomials. Typically, cell is an entry from the result of the function LeftCells.

    gap> v := X( Cyclotomics ) ;; v.name := "v";;
    gap> W := Hecke(CoxeterGroup( "H", 3), v^2, v );
    Hecke(CoxeterGroup("H", 3),[ v^2, v^2, v^2 ],[ v, v, v ])
    gap> c := LeftCells( CoxeterGroup( W ) );;
    gap> List( c, i -> Length( i[ 1 ] ) );
    [ 1, 6, 5, 8, 5, 6, 1, 5, 8, 5, 5, 6, 6, 5, 8, 5, 5, 8, 5, 6, 6, 5 ]
    gap> LeftCellRepresentation(W,c[3]);
    [ [ [ -v^0, v, 0*v^0, 0*v^0, 0*v^0 ], 
          [ 0*v^0, v^2, 0*v^0, 0*v^0, 0*v^0 ], 
          [ 0*v^0, v, -v^0, v, 0*v^0 ], 
          [ 0*v^0, 0*v^0, 0*v^0, v^2, 0*v^0 ], 
          [ 0*v^0, 0*v^0, 0*v^0, 0*v^0, v^2 ] ], 
      [ [ v^2, 0*v^0, 0*v^0, 0*v^0, 0*v^0 ], [ v, -v^0, v, 0*v^0, 0*v^0 ],
          [ 0*v^0, 0*v^0, v^2, 0*v^0, 0*v^0 ], 
          [ 0*v^0, 0*v^0, v, -v^0, v ], 
          [ 0*v^0, 0*v^0, 0*v^0, 0*v^0, v^2 ] ], 
      [ [ -v^0, v, 0*v^0, 0*v^0, 0*v^0 ], 
          [ 0*v^0, v^2, 0*v^0, 0*v^0, 0*v^0 ], 
          [ 0*v^0, 0*v^0, v^2, 0*v^0, 0*v^0 ], 
          [ 0*v^0, 0*v^0, 0*v^0, v^2, 0*v^0 ], 
          [ 0*v^0, 0*v^0, 0*v^0, v, -v^0 ] ] ]

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

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