CharTable( H )
CharTable
returns the character table record of the Iwahori-Hecke
algebra H. This is basically the same as the character table of a
Coxeter group described earlier with the exception that the component
irreducibles
contains the matrix of the values of the irreducible
characters of the generic Iwahori-Hecke algebra specialized at the
parameters in the component parameter
of H. Thus, if all these
parameters are equal to~1 in QQ then the component irreducibles
just contains the ordinary character table of the underlying Coxeter
group.
The function CharTable
first recognizes the type of H, then calls
special functions for each type involved in H and finally forms the
direct product of all these tables.
gap> W := CoxeterGroup( "G", 2 );; gap> u := X( Rationals );; u.name := "u";; gap> v := X( LaurentPolynomialRing( Rationals ) );; v.name := "v";; gap> u := u * v^0;; gap> H := Hecke( W, [ u^2, v^2 ], [ u, v ] ); Hecke(CoxeterGroup("G", 2),[ u^2*v^0, v^2 ],[ u*v^0, v ]) gap> Display( CharTable( H ) ); H(G2) 2 2 2 2 1 1 2 3 1 . . 1 1 1 ~A_1 A_1 G_2 A_2 A_1 + ~A_1 2P A_2 A_2 3P ~A_1 A_1 A_1 + ~A_1 A_1 + ~A_1 phi_{1,0} 1 v^2 (u^2) (u^2)v^2 (u^4)v^4 (u^6)v^6 phi_{1,6} 1 -1 -1 1 1 1 phi_{1,3}' 1 v^2 -1 -v^2 v^4 -v^6 phi_{1,3}'' 1 -1 (u^2) (-u^2) (u^4) (-u^6) phi_{2,1} (2) v^2+(-1) (u^2-1) (u)v (-u^2)v^2 (-2u^3)v^3 phi_{2,2} (2) v^2+(-1) (u^2-1) (-u)v (-u^2)v^2 (2u^3)v^3
As mentioned before, the record components classparam
, classnames
and
irredinfo
contain canonical labels and parameters for the classes and
Character tables for Coxeter groups and also ChevieCharInfo). For direct products,
sequences of such canonical labels of the individual types are given.
We can also have character tables for algebras where the parameters are not necessarily indeterminates:
gap> H1 := Hecke( W, [ E(6)^2, E(6)^4 ],[ E(6), E(6)^2 ] ); Hecke(CoxeterGroup("G", 2),[ E(3), E(3)^2 ],[ -E(3)^2, E(3) ]) gap> ct := CharTable( H1 ); CharTable( "H(G2)" ) gap> Display( ct ); H(G2) 2 2 2 2 1 1 2 3 1 . . 1 1 1 ~A_1 A_1 G_2 A_2 A_1 + ~A_1 2P A_2 A_2 3P ~A_1 A_1 A_1 + ~A_1 A_1 + ~A_1 phi_{1,0} 1 A /A 1 1 1 phi_{1,6} 1 -1 -1 1 1 1 phi_{1,3}' 1 A -1 -A /A -1 phi_{1,3}'' 1 -1 /A -/A A -1 phi_{2,1} 2 B /B -1 -1 2 phi_{2,2} 2 B /B 1 -1 -2 A = E(3)^2 = (-1-ER(-3))/2 = -1-b3 B = E(3)+2*E(3)^2 = (-3-ER(-3))/2 = -2-b3 gap> RankMat( ct.irreducibles ); 5
The last result tells us that the specialized character table is no more invertible.
Character tables of Iwahori--Hecke algebras were introduced in GP93; see also the introduction to this chapter for further information about the origin of the various tables.
This function requires the package "chevie" (see RequirePackage).
GAP 3.4.4