83.2 CriticalPair

CriticalPair( W, y, w, ly )

Given an element y of length ly in the Coxeter group W and an element w the function CriticalPair returns a triple (z,w,lz) where (z,w) is a critical pair (i.e., we have {mathcal{L}}(w) subseteq {mathcal{L}}(z) and {mathcal{R}}(w) subseteq {mathcal{R}}(z) and lz is the length of z. This critical pair is chosen so that the corresponding Kazhdan--Lusztig polynomials P_{z,w} and P_{y,w} are equal.

    gap> W := CoxeterGroup( "F", 4 );
    CoxeterGroup("F", 4)
    gap> w := LongestCoxeterElement( W ) * W.generators[1];;
    gap> CoxeterLength( W, w );
    23
    gap> y := PermCoxeterWord( W, [ 1, 2, 3, 4 ] );;
    gap> cr := CriticalPair( W, y, w, 4 );;
    gap> CoxeterWord( W, cr[1] );
    [ 2, 3, 2, 1, 3, 4, 3, 2, 1, 3, 2, 3, 4, 3, 2, 3 ]
    gap> cr[3];
    16
    gap> KazhdanLusztigPolynomial( W, y, w, 4, 23 );
    q^3 + 1
    gap> KazhdanLusztigPolynomial( W, cr[1], cr[2], 16, 23 );
    q^3 + 1 

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

Previous Up Top Next
Index