[GAP Forum] Use GAP3 program ( especially the package CHIEVE) for computing the h-polynomials of Lusztig

[Jean Michel]

Hello,

> I know about the computation of Kazhdan-Lusztig basis and change between "T" and "C"-basis using GAP3. But how can I compute the $h$-polynomials as defined in
> $$c_xc_y = \sum_{z \in W}h_{x,y,z}c_z,$$
> for any $x,y,z$ in a Coxeter group $W$, using GAP3?
> Or, how can I compute the $f$-polynomials as defined in
> $$T_xT_y = \sum_{z \in W}f_{x,y,z}T_z.$$
> using GAP3?

It is straightforward to do what you want:

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)

Best regards,
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Jean MICHEL, Groupes et representations, IMJ-PRG UMR7586 tel.(33)157279144
Bureau 639 Bat. Sophie Germain Case 7012 - 75205 PARIS Cedex 13