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.

- ComplexReflectionGroup
- Operations for complex reflection groups
- Hecke for complex reflection groups
- Operations for cyclotomic Hecke algebras

`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*).

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).

`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).

`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).
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GAP 3.4.4

April 1997