Use GAP3 program for computing the h-polynomials of Lusztig

Question

I am using GAP to compute things in Kazhdan-Lusztig theory, especially using the package "Chevie". According to the GAP3 Manual, the base change between the "C"-basis and "T"-basis can be computed, for example, from

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)

we see that $c_1 = -vT_e+v^{-1}T_1$, and $T_1 = v^2c_e + vc_1$.

For any $x,y$ and $z$ in a Coxeter group $W$, the $h$-polynomials are defined in $$c_xc_y = \sum_{z \in W}h_{x,y,z}c_z.$$

My problem is

how can I compute the $h$-polynomials with GAP3 program?

Answer 1

This is the answer in the GAP Forum mail list, given by Jean Michel

gap> W:=CoxeterGroup("A",3);q:=X(Rationals);q.name:="q";
CoxeterGroup("A",3)
X(Rationals)
"q"
gap> H:=Hecke(W,q^2,q);C:=Basis(H,"C");;T:=Basis(H,"T");;
Hecke(A3,q^2,q)
gap> a:=T(1,2)^2;
q^2T(2,1)+(q^2-1)T(1,2,1)
gap> a.coeff;
[ q^2, q^2 - 1 ]
gap> Coefficient(a,[2,1]);
q^2
gap> Coefficient(a,[1,2,1]);
q^2 - 1
gap> b:=C(1,2)^2;
C(1,2)+(-q-q^-1)C(1,2,1)
gap> b.coeff;
[ -q - q^(-1), q^0 ]
gap> Coefficient(b,[1,2,1]);
-q - q^(-1)

Thanks to @AlexanderKonovalov for reminding me of answering my own question after knowing the answer to it.