84.8 PhiFactors

PhiFactors( WF )

Let W be the Coxeter group corresponding to the Coxeter coset WF, and let V be the vector space of dimension W.rank on which W acts as a reflection group. Let f_1,ldots,f_n be the basic invariants of W on the symmetric algebra SV of V. The matrix WF.F0Mat has the f_i as eigenvectors. The corresponding eigenvalues, sorted in order of increasing degrees of the f_i are called the factors of F_0 acting on V.

    gap> W := CoxeterGroup( "E", 6 );; WF := CoxeterCoset( W );
    CoxeterCoset(CoxeterGroup("E", 6))
    gap> phi := PermCoxeterWord( W, 
    >       [ 6, 5, 4, 2, 3, 1, 4, 3, 5, 4, 2, 6, 5, 4, 3, 1 ] );;
    gap> HF := CoxeterSubCoset( WF, [ 2..5 ], phi );;
    gap> PrintDynkinDiagram( HF );
    phi acts as (2,3,5) on the component below
    D4   2
          \
           4 - 5
          /
         3
    gap> PhiFactors( HF );
    [ E(3), E(3)^2, 1, E(3), E(3)^2, 1 ]
    gap> ReflectionDegrees( CoxeterGroup( HF ) );
    [ 1, 1, 2, 4, 4, 6 ] 

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

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