83.9 Hecke elements of the $D$ basis

Basis( H, "D" )

returns a function which gives the D-basis of the (one parameter generic) Iwahori-Hecke algebra H (see cite[(5.1)]Lus85). This can be defined by [ D_x := v^-NC_xw_0^prime T_w_0 mbox for every x in W, ] where N denotes the number of positive roots in the root system of W and w_0 is the longest element of W. The D-basis is dual to the C-basis with respect to the non-degenerate form H times H rightarrow {ZZ}[v,v^{-1}], (h_1,h_2) mapsto tau(h_1 cdot h_2) where tau colon H rightarrow {ZZ}[v,v^{-1}] is the linear form such that tau(T_1)=1 and tau(T_x)=0 for x neq 1. We have D_x=beta(C_{w_0x}^prime) for all x in W (see BetaInvolution in section Operations for Hecke elements of the $T$ basis).

    gap> W := CoxeterGroup( "B", 2 );;
    gap> v := X( Rationals );; v.name := "v";;
    gap> H := Hecke( W, v^2, v );
    Hecke(CoxeterGroup("B", 2),[ v^2, v^2 ],[ v, v ])
    gap> T := Basis( H, "T" );
    function ( arg ) ... end
    gap> D := Basis( H, "D" );
    function ( arg ) ... end
    gap> D( T( 1 ) );
    vD(1)-v^2D(1,2)-v^2D(2,1)+v^3D(1,2,1)+v^3D(2,1,2)-v^4D(1,2,1,2)
    gap> BetaInvolution( D( 1 ) );
    C'(2,1,2) 

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

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