78.2 Functions for reflection subgroups

All functions for Coxeter groups are actually defined for reflection subgroups. The generators for the subgroups are labeled according to the corresponding number of the root they represent in the parent group. This affects the labeling given by all functions dealing with words and generators, e.g., PrintDynkinDiagram or PermCoxeterWord.

    gap> W := CoxeterGroup( "F", 4 );          
    CoxeterGroup("F", 4)
    gap> H := ReflectionSubgroup( W, [ 10, 11, 12 ] );
    ReflectionSubgroup(CoxeterGroup("F", 4), [ 10, 11, 12 ])
    gap> PrintDynkinDiagram( H );
    C2    11 > 10
    ~A1    12
    gap> LongestCoxeterWord( H );
    [ 10, 11, 10, 11, 12 ] 

Also, as one may notice in the example above, there is one particularity of the functions CartanType, CartanName and PrintDynkinDiagram for Coxeter subgroups: an irreducible subsystem which consists of short roots in a system which has longer roots (i.e., type "B", "C", "G" or "F") is labeled as type "~A".

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