75.6 Operations and functions for Coxeter groups

All permutation group operations are defined on Coxeter groups. However, the following operations and functions have been rewritten or added, respectively, to take advantage of the particular structure of real reflection groups:

=:

Two Coxeter data are equal if they are equal as permutation groups and the fields simpleRoots and simpleCoroots agree (independently of the value of any other bound fields). Also the fields .omega should be equal (if the field is absent --- omega not specified --- this is considered specifying the trivial group).

Print:

prints a Coxeter group in a form that can be input back in GAP as a Coxeter group.

Size:

uses the classification of Coxeter groups to work faster (specifically, uses the function ReflectionDegrees).

Elements:

returns the set of elements. They are computed using CoxeterElementsLength. (Note that in an earlier version of the package the elements were sorted by length. You can still get such a list by Concatenation( List( [1..W.N], i - CoxeterElementsLength(W, i))))

CartanType:

returns CartanType of the Cartan matrix.

PrintDynkinDiagram:

prints the Dynkin diagram corresponding to the Cartan matrix.

CartanName:

returns the CartanName of the CartanType.

ConjugacyClasses:

Uses classification of Coxeter groups to work faster, and the resulting list is given in the same order as the result of ChevieClassInfo (see ChevieClassInfo).

CharTable:

Uses the classification of Coxeter groups to work faster, and the result has better labeling than the default (see Chapter Character tables for Coxeter groups).

ChevieClassInfo:

Is part of the Coxeter groups operations record in order to have versions for Coxeter groups and Coxeter cosets. This function returns additional information on the classes which is contained in the character table, but this function returns it without first computing CharTable(W). See the explicit description in ChevieClassInfo.

CharParams:

Is part of the Coxeter groups operations record in order to have versions for Coxeter groups and Coxeter cosets. This function returns the list of parameters for the irreducible characters of W: partitions for type A, double partitions for type B, etc... This is used by functions which return information for individual characters, like FakeDegree, SchurElement, etc...

CharName:

Is part of the Coxeter groups operations record in order to have versions for Coxeter groups and Coxeter cosets. This function takes as argument a parameter for a character (see CharParams) and returns a string which is used to label the character is various displayed tables.

ChevieCharInfo:

This function returns additional information on the irreducible characters, see ChevieCharInfo for more details.

PositionClass, ClassInvariants, FusionConjugacyClasses:

Use the classification of Coxeter groups to work faster. PositionClass(W,x) returns the index of the conjugacy class of x in the list of the classes of W, see PositionClass.

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