83.1 KazhdanLusztigPolynomial

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

returns the Kazhdan-Lusztig polynomial in the indeterminate X(Rationals) corresponding to the elements y and w (given as permutations) of the Coxeter group W. The optional variables ly and lw contain the length of y and w, respectively. The above form for the input has been chosen for efficiency reasons. If one prefers to give as input just two reduced expressions, one can define a new function as follows (for example):

    gap> klpol := function( W, x, y) 
    >      return KazhdanLusztigPolynomial( W, PermCoxeterWord( W, x ),
    >                  PermCoxeterWord( W, y ), Length( x ), Length( y ) );
    >    end;
    function ( W, x, y ) ... end 

We use this function in the following example where we compute the polynomials P_{1,w} for all elements w in the Coxeter group of type A_3.

    gap> q := X( Rationals );; q.name := "q";;
    gap> W := CoxeterGroup( "B", 3 );;
    gap> el := CoxeterWords( W );
    [ [  ], [ 3 ], [ 2 ], [ 1 ], [ 3, 2 ], [ 2, 1 ], [ 2, 3 ], [ 1, 3 ], 
      [ 1, 2 ], [ 2, 1, 2 ], [ 3, 2, 1 ], [ 2, 3, 2 ], [ 2, 1, 3 ], 
      [ 1, 2, 1 ], [ 1, 3, 2 ], [ 1, 2, 3 ], [ 3, 2, 1, 2 ], 
      [ 2, 1, 2, 3 ], [ 2, 3, 2, 1 ], [ 2, 1, 3, 2 ], [ 1, 2, 1, 2 ], 
      [ 1, 3, 2, 1 ], [ 1, 2, 1, 3 ], [ 1, 2, 3, 2 ], [ 3, 2, 1, 2, 3 ], 
      [ 2, 1, 2, 3, 2 ], [ 2, 3, 2, 1, 2 ], [ 2, 1, 3, 2, 1 ], 
      [ 1, 3, 2, 1, 2 ], [ 1, 2, 1, 2, 3 ], [ 1, 2, 1, 3, 2 ], 
      [ 1, 2, 3, 2, 1 ], [ 2, 3, 2, 1, 2, 3 ], [ 2, 1, 2, 3, 2, 1 ], 
      [ 2, 1, 3, 2, 1, 2 ], [ 1, 3, 2, 1, 2, 3 ], [ 1, 2, 1, 2, 3, 2 ], 
      [ 1, 2, 1, 3, 2, 1 ], [ 1, 2, 3, 2, 1, 2 ], [ 2, 1, 2, 3, 2, 1, 2 ],
      [ 2, 1, 3, 2, 1, 2, 3 ], [ 1, 2, 3, 2, 1, 2, 3 ], 
      [ 1, 2, 1, 2, 3, 2, 1 ], [ 1, 2, 1, 3, 2, 1, 2 ], 
      [ 2, 1, 2, 3, 2, 1, 2, 3 ], [ 1, 2, 1, 2, 3, 2, 1, 2 ], 
      [ 1, 2, 1, 3, 2, 1, 2, 3 ], [ 1, 2, 1, 2, 3, 2, 1, 2, 3 ] ]
    gap> List( el, w -> klpol( W, [], w ) );
    [ q^0, q^0, q^0, q^0, q^0, q^0, q^0, q^0, q^0, q^0, q^0, q^0, q^0, 
      q^0, q^0, q^0, q^0, q^0, q^0, q + 1, q^0, q^0, q^0, q^0, q + 1, 
      q^0, q^0, q + 1, q^0, q^0, q + 1, q + 1, q^0, q + 1, q^0, q + 1, 
      q^0, q^2 + 1, q + 1, q^2 + q + 1, q + 1, q + 1, q^0, q^0, q^2 + 1, 
      q^0, q + 1, q^0 ] 

The set of Kazhdan--Lusztig polynomials that were found during this computation is contained in the record component klpol (as lists of coefficients):

    gap> W.klpol;
    [ [ 1, 1 ], [ 1 ], [ 1, 0, 1 ], [ 1, 1, 1 ] ] 

Thus, we have found four different Kazhdan-Lusztig polynomials, namely 1+q, 1, 1+q^2 and 1+q+q^2.

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

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GAP 3.4.4
April 1997