GoodCoxeterWord( W, w )
Let W be a Coxeter group with associated braid monoid B^+.
GoodCoxeterWord checks if the element w of W (given as sequence of
generators of W) represents a ``good element'' in the sense of
Geck-Michel GM97 of the braid monoid, i.e., if bw^d (where d
is the order of the element w in W, and bw is the element of
B^+_{text{red}} with image w) is a product of (the braid elements
corresponding to) longest elements in a decreasing chain of parabolic
subgroups of W. If this is true, then a list of couples, the
corresponding subsets of the generators with their multiplicities in the
chain, is returned. Otherwise, false is returned.
Good elements have nice properties with respect to their eigenvalues in
irreducible represen-tations of the Hecke-Iwahori algebra associated to
W. The representatives in the component classtext of
ChevieClassInfo(W) are all good elements of minimal length in their
class.
gap> W := CoxeterGroup( "F", 4 );;
gap> w:=[ 2, 3, 2, 3, 4, 3, 2, 1, 3, 4 ];;
gap> GoodCoxeterWord( W, w );
[ [ [ 1 .. 4 ], 2 ], [ [ 3, 4 ], 4 ] ]
gap> OrderPerm( PermCoxeterWord( W, w ) );
6
gap> Braid( W )( w ) ^ 6;
w0^2.343.343.343.343
gap> GoodCoxeterWord( W, [ 3, 2, 3, 4, 3, 2, 1, 3, 4, 2 ] );
false
This function requires the package "chevie" (see RequirePackage).
GAP 3.4.4