80 Complex reflection groups, cyclotomic algebras

Preliminary support for complex reflection groups and cyclotomic Hecke algebras has been added to the CHEVIE package. A complex reflection group is a group W acting on a vector space V, and generated by pseudo-reflections in V. The field of definition of W is defined to be the field of definition of V. It turns out that, as for rational reflection groups (Weyl groups), all representations of a complex reflection group W are defined over the field of definition of W (cf. Ben76 and D.~Bessis thesis). Similarly to Coxeter groups, complex reflection groups are represented by the permutation representation on a set of roots in V invariant by W and such that all reflections in W are reflections with respect to some root (see ComplexReflectionGroup). However there is no general theory on how to construct a nice set of roots for an arbitrary reflection group; the roots given in GAP where obtained case-by-case in an ad hoc way.

Irreducible complex reflection groups have been classified by Shephard and Todd. They contain one infinite family depending on 3 parameters, and 34 ``exceptional'' groups (which have been given by Shephard and Todd a number which actually varies from 4 to 37, and covers also the exceptional Coxeter groups, e.g., CoxeterGroup("E",8) is the group of Shephard-Todd number 37).

The cyclotomic Hecke algebra (see Hecke for complex reflection groups) corresponding to a complex reflection group is defined in a similar way as the Iwahori--Hecke algebra; for details see BM93. G.~Malle has computed character tables for some of these algebras, including all 2-dimensional groups, see BM93 and Mal96; CHEVIE contains 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.

Subsections

  1. ComplexReflectionGroup
  2. Operations for complex reflection groups
  3. Hecke for complex reflection groups
  4. Operations for cyclotomic Hecke algebras

80.1 ComplexReflectionGroup

ComplexReflectionGroup( STnumber )

ComplexReflectionGroup( p, q, r )

The first form of ComplexReflectionGroup returns the complex reflection group which has Shephard-Todd number STnumber, see ST54. The second form returns the imprimitive complex reflection group G(p,q,r).

    gap> G := ComplexReflectionGroup( 4 );
    ComplexReflectionGroup(4)
    gap> ReflectionDegrees( G );
    [ 4, 6 ]
    gap> Size( G );
    24
    gap> q := X( Cyclotomics );; q.name := "q";;
    gap> FakeDegrees( G, q );
    [ q^0, q^8, q^4, q^7 + q^5, q^3 + q, q^5 + q^3, q^6 + q^4 + q^2 ] 

Complex reflection groups are represented as permutation group records

with the following additional fields:

roots:

a set of complex roots in V, given as a matrix, on which W has a faithful permutation representation. roots[1..semisimpleRank] should be linearly independent. Roots are not always of same length, and sometimes the number of roots may be greater than the order of W!

semisimpleRank:

the dimension of the subspace of V generated by the roots (for an irreducible group, equal to the dimension of V).

80.2 Operations for complex reflection groups

All permutation group operations are defined on complex reflection groups. The following operations and functions have been rewritten to take advantage of the particular structure of reflection groups:

Print:

prints a complex reflection group in a form that can be input back in GAP.

ReflectionDegrees:

the list of reflection degrees.

Size:

Uses the product of the ReflectionDegrees to work faster.

FakeDegrees(W,q):

The list of fake degrees as polynomials in q.

ReflectionCharValue(W,w):

The value of the reflection character on the element w of W, given as a permutation of the roots.

These functions require the package "chevie" (see RequirePackage).

80.3 Hecke for complex reflection groups

Hecke( G, para )

returns the cyclotomic Hecke algebra corresponding to the complex reflection group G. The parameters of this algebra are specified in the variable para, which may be either a single value or a list of parameters.

    gap> G := ComplexReflectionGroup( 4, 2, 3 );
    ComplexReflectionGroup(4,2,3)
    gap> v := X( Cyclotomics );; v.name := "v";;
    gap> CH := Hecke( G, v );
    Hecke(ComplexReflectionGroup(4,2,3),v) 

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

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 Next
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GAP 3.4.4
April 1997