80.4 Operations for cyclotomic Hecke algebras

Group:

returns the complex reflection group from which the cyclotomic Hecke algebra was generated.

Print:

prints the cyclotomic Hecke algebra in a form which can be read back into GAP.

SchurElements:

returns Schur elements (analogously defined as for Iwahori--Hecke algebras) for some types of exceptional cyclotomic Hecke algebras.

CharTable:

returns the character table for some types of cyclotomic Hecke algebras, namely those of type G(e,1,1), G_4, G_5, G_6, G_8, G_9, G_{12} and G_{25} in the Shephard-Todd classification. This is a record with exactly the same components as for the corresponding complex reflection group but where the component irreducibles contains the values of the irreducible characters of the algebra on certain basis elements T_w where w runs over the elements in the component classtext. Thus, the value are now polynomials in the parameters of the algebra.

    gap> G := ComplexReflectionGroup( 4 );
    ComplexReflectionGroup(4)
    gap> v := X( Cyclotomics );; v.name := "v";;
    gap> CH := Hecke( G, v );
    Hecke(ComplexReflectionGroup(4),v)
    gap> Display( CharTable( CH ) );
    H(G4)

           2  3       3          1            1   2         1           1
           3  1       1          1            1   .         1           1

             1a      2a         3a           3b  4a        6a          6b
          2P 1a      1a         3b           3a  2a        3a          3b
          3P 1a      2a         1a           1a  4a        2a          2a

phi_{1,0}     1     v^6          v          v^2 v^3       v^2        v^10
phi_{1,4}     1       1          A           /A   1        /A           A
phi_{1,8}     1       1         /A            A   1         A          /A
phi_{2,1}     2 (-2)v^3   v+(E(3)) v^2+(E(3)^2)   .   (E(3))v (E(3)^2)v^5
phi_{2,3}     2 (-2)v^3 v+(E(3)^2)   v^2+(E(3))   . (E(3)^2)v   (E(3))v^5
phi_{2,5}     2      -2         -1           -1   .         1           1
phi_{3,2}     3  (3)v^2        v-1        v^2-1  -v         .           .

A = E(3)
  = (-1+ER(-3))/2 = b3 

This function requires the package "chevie" (see RequirePackage). Previous Up Top
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