78.3 ReducedInCoxeterCoset

ReducedInCoxeterCoset( W , w )

w is an automorphism of the Coxeter group W or of a parent group of W, given as a permutation of the roots. ReducedInCoxeterCoset returns the unique element in the right coset W.w which sends all roots of W to positive roots.

    gap> W := CoxeterGroup("F", 4 );;
    gap> H := ReflectionSubgroup( W, [ 1, 2, 9, 16 ] );;
    gap> PrintDynkinDiagram( H );
    D4   9
          \       
           1 - 16
          /
         2
    gap> w := PermCoxeterWord( W, [ 3, 2, 3, 4, 3, 2 ] );;
    gap> f := ReducedInCoxeterCoset( H, w );;
    gap> CoxeterWord( W, f );
    [ 4, 3 ]
    gap> H.rootInclusion{[ 1 ..4 ]};
    [ 1, 2, 9, 16 ]

The triality automorphism of D_4:

    gap> OnTuples( H.rootInclusion{[ 1 .. 4 ]}, f );
    [ 1, 9, 16, 2 ]

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