84.4 CartanType for Coxeter cosets

CartanType( WF )

returns the type of the Coxeter coset WF. This consists of a list of records, one for each orbit of WF.phi on the irreducible components of the Dynkin diagram of CoxeterGroup(WF), which have two fields:

orbit:

is a list of types of the irreducible components in the orbit. These types are the same as returned by the function CartanType for an irreducible untwisted Coxeter group (see CartanType in chapter Root systems and finite Coxeter groups): a couple [type,indices] (a triple for type I_2(n)). The components are ordered according to the action of WF.phi, so WF.phi maps the generating permutations with indices in the first type to indices in the second type in the same order as stored in the type, etc ldots
phi:
if k is the number of irreducible components in the orbit, this is the permutation which describes the action of WF.phi^k on the simple roots of the first irreducible component in the orbit.

    gap> W := CoxeterCoset( CoxeterGroup( "A", 2, "A", 2 ), (1,3,2,4) );
    CoxeterCoset(CoxeterGroup("A", 2, "A", 2), (1,3,2,4))
    gap> CartanType( W );
    [ rec(
          orbit := [ [ "A", [ 1, 2 ] ], [ "A", [ 3, 4 ] ] ],
          phi := (1,2) ) ]

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

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