83.3 KazhdanLusztigCoefficient

KazhdanLusztigCoefficient( W, y, w, [ ly, lw,] k )

returns the k-th coefficient of the Kazhdan-Lusztig polynomial corresponding to the elements y and w, which must be given as permutations on the root vectors, of the Coxeter group W. Again, the optional variables ly and lw contain the length of y and w, respectively.

    gap> W := CoxeterGroup( "B", 4 );;
    gap> y := [ 1, 2, 3, 4, 3, 2, 1 ];;
    gap> py := PermCoxeterWord( W, y );
    ( 1,28)( 2,15)( 4,27)( 6,16)( 7,24)( 8,23)(11,20)(12,17)(14,30)(18,31)
    (22,32)
    gap> x := [ 1 ];;
    gap> px := PermCoxeterWord( W, x );
    ( 1,17)( 2, 8)( 6,11)(10,14)(18,24)(22,27)(26,30)
    gap> Bruhat( W, px, py );
    true
    gap> List([0..3],i->KazhdanLusztigCoefficient( W, px, py, 1, 7, i ) );
    [ 1, 2, 1, 0 ] 

So the Kazhdan-Lusztig polynomial corresponding to x and y is 1+2u+u^2.

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

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