76.15 Bruhat

Bruhat( W, y, w[, ly, lw] )

returns true, if the element y is less than or equal to the element w of the Coxeter group W for the Bruhat order, and false otherwise (y is less than w if a reduced expression for y can be extracted from one for w). Both y and w must be given as permutations on the root vectors of W. The optional arguments ly, lw can contain the length of the elements y and w. (In a computation with many calls to Bruhat this may speed up the computation considerably.) See cite[(5.9) and (5.10)]Hum90 for further properties of the Bruhat order.

    gap> W := CoxeterGroup( "H", 3 );;
    gap> w := PermCoxeterWord( W, [ 1, 2, 1, 3 ] );;
    gap> b := Filtered( Elements( W ), i -> Bruhat( W, i, w, 
    >                                 CoxeterLength( W, i ), 4 ) );;
    gap> List( b, x -> CoxeterWord( W, x ) );
    [ [  ], [ 3 ], [ 2 ], [ 2, 1 ], [ 2, 3 ], [ 2, 1, 3 ], [ 1 ], 
      [ 1, 3 ], [ 1, 2 ], [ 1, 2, 1 ], [ 1, 2, 3 ], [ 1, 2, 1, 3 ] ] 

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

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