76.16 MatXPerm

MatXPerm( W, w )

Let w be a permutation of the roots of the Coxeter datum W acting on the vector space V. MatXPerm returns the matrix of a linear transformation of V which acts trivially on the orthogonal of the coroots and has same effect as w on the simple roots. Only the image of the simple roots by w is used.

    gap> W := CoxeterGroup( 
    > [ [ 2, 0,-1, 0, 0, 0, 1 ], [ 0, 2, 0,-1, 0, 0, 0 ],
    >   [-1, 0, 2,-1, 0, 0,-1 ], [ 0,-1,-1, 2,-1, 0, 0 ],
    >   [ 0, 0, 0,-1, 2,-1, 0 ], [ 0, 0, 0, 0,-1, 2, 0 ] ],
    > [ [ 1, 0, 0, 0, 0, 0, 0 ], [ 0, 1, 0, 0, 0, 0, 0 ],
    >   [ 0, 0, 1, 0, 0, 0, 0 ], [ 0, 0, 0, 1, 0, 0, 0 ],
    >   [ 0, 0, 0, 0, 1, 0, 0 ], [ 0, 0, 0, 0, 0, 1, 0 ] ] );;
    gap> w0 := LongestCoxeterElement( W );;
    gap> mx := MatXPerm( W, w0 );
    [ [ 0, 0, 0, 0, 0, -1, 1 ], [ 0, -1, 0, 0, 0, 0, 2 ], 
      [ 0, 0, 0, 0, -1, 0, 3 ], [ 0, 0, 0, -1, 0, 0, 4 ], 
      [ 0, 0, -1, 0, 0, 0, 3 ], [ -1, 0, 0, 0, 0, 0, 1 ], 
      [ 0, 0, 0, 0, 0, 0, 1 ] ] 

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

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