Basis( H, "D'" )
returns a function which gives the D^prime-basis of the (one parameter
generic) Iwahori-Hecke algebra H (see cite[(5.1)]Lus85). This can be
defined by
[
D_x^prime := v^-NC_xw_0 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^prime-basis is basis
dual to the C^prime-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^prime=Alt(D_x) for all x in W (see AltInvolution
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> Dp := Basis( H, "D'" ); function ( arg ) ... end gap> AltInvolution( Dp( 1 ) ); D(1) gap> Dp( 1 )^3; (v+2v^-1-5v^-5-9v^-7-8v^-9-4v^-11-v^-13)D'()+(v^2+2+v^-2)D'(1)
This function requires the package "chevie" (see RequirePackage).
GAP 3.4.4