83.7 Hecke elements of the $C$ basis

Basis( H, "C" )

returns a function which gives the C-basis of the (one parameter generic) Iwahori-Hecke algebra H. This is defined as follows (see cite[(5.1)]Lus85, for example). Let W be the underlying finite Coxeter group. For x,y in W let P_{x,y} be the corresponding Kazhdan--Lusztig polynomial. If {T_w mid w in W} denotes the usual T-basis and u=v^2 the parameter for H, then [ C_x:=sum_y leq x (-1)^l(x)-l(y)P_x,y(v^-2)v^l(x)-2l(y) T_y quad mbox for every x in W.] For example, we have C_s=v^{-1}T_s-vT_1 for s in S. Thus, the transformation matrix between the T-basis and the C-basis is lower unitriangular, with powers of v along the diagonal. The multiplication rules for this new basis are given as follows. [ C_s cdot C_x =left{ beginarrayll -(v+v^-1)C_x & mbox, if sx
C_sx+sum_y mu(y,x)C_y & mbox, if sx>xendarrayright.] where the sum is over all y such that y, l(y) notequiv l(x)~mod~2 and sy. The coefficient mu(y,x) is the coefficient of degree (l(x)-l(y)-1)/2 in the Kazhdan--Lusztig polynomial P_{x,y}.

    gap> W := CoxeterGroup( "B", 3 );;
    gap> v := X( Rationals );; v.name := "v";;
    gap> H := Hecke( W, v^2, v );
    Hecke(CoxeterGroup("B", 3),[ v^2, v^2, v^2 ],[ v, v, v ])
    gap> T := Basis( H, "T" );
    function ( arg ) ... end
    gap> C := Basis( H, "C" );
    function ( arg ) ... end
    gap> T( C( 1 ) );
    -vT()+v^-1T(1)
    gap> C( T( 1 ) );
    v^2C()+vC(1) 

    gap> ref := HeckeReflectionRepresentation( H );;
    gap> c := CharRepresentationWords( ref, CoxeterConjugacyClasses( W ) );
    [ 3, 2*v^2 - 1, v^8 - 2*v^4, -3*v^12, 2*v^2 - 1, v^4, v^4 - 2*v^2, 
      -v^6, v^4 - v^2, 0*v^0 ]
    gap> List( ChevieClassInfo( W ).classtext, i -> 
    >                             HeckeCharValues( C( i ), c ) );
    [ 3*v^0, -v - v^(-1), 0*v^0, 0*v^0, -v - v^(-1), 2*v^0, 0*v^0, 0*v^0, 
      v^0, 0*v^0 ]