82.3 CharTable for Hecke algebras

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).

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