Kazhdan-Lusztig Polynomials

AUTHORS:

  • Daniel Bump (2008): initial version
  • Alan J.X. Guo (2014-03-18): R_tilde() method.
sage.combinat.kazhdan_lusztig.KazhdanLusztigPolynomial

A Kazhdan-Lusztig polynomial.

INPUT:

  • W – a Weyl Group
  • q – an indeterminate

OPTIONAL:

  • trace – if True, then this displays the trace: the intermediate results. This is instructive and fun.

The parent of q may be a PolynomialRing or a LaurentPolynomialRing.

EXAMPLES:

sage: W = WeylGroup("B3",prefix="s")
sage: [s1,s2,s3] = W.simple_reflections()
sage: R.<q> = LaurentPolynomialRing(QQ)
sage: KL = KazhdanLusztigPolynomial(W,q)
sage: KL.P(s2,s3*s2*s3*s1*s2)
1 + q

A faster implementation (using the optional package Coxeter 3) is given by:

sage: W = CoxeterGroup(['B', 3], implementation='coxeter3') # optional - coxeter3
sage: W.kazhdan_lusztig_polynomial([2], [3,2,3,1,2])        # optional - coxeter3
q + 1