Kazhdan-Lusztig Polynomials

AUTHORS:

  • Daniel Bump (2008): initial version
  • Alan J.X. Guo (2014-03-18): R_tilde() method.
class sage.combinat.kazhdan_lusztig.KazhdanLusztigPolynomial(W, q, trace=False)

Bases: sage.structure.unique_representation.UniqueRepresentation, sage.structure.sage_object.SageObject

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.

REFERENCES:

[KL79]D. Kazhdan and G. Lusztig. Representations of Coxeter groups and Hecke algebras. Invent. Math. 53 (1979). no. 2, 165–184. doi:10.1007/BF01390031 MathSciNet MR0560412
[Dy93]M. J. Dyer. Hecke algebras and shellings of Bruhat intervals. Compositio Mathematica, 1993, 89(1): 91-115.
[BB05]A. Bjorner, F. Brenti. Combinatorics of Coxeter groups. New York: Springer, 2005.

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
1 + q
P(x, y)

Return the Kazhdan-Lusztig \(P\) polynomial.

If the rank is large, this runs slowly at first but speeds up as you do repeated calculations due to the caching.

INPUT:

  • x, y – elements of the underlying Coxeter group

See also

kazhdan_lusztig_polynomial for a faster implementation using Fokko Ducloux’s Coxeter3 C++ library.

EXAMPLES:

sage: R.<q> = QQ[]
sage: W = WeylGroup("A3", prefix="s")
sage: [s1,s2,s3] = W.simple_reflections()
sage: KL = KazhdanLusztigPolynomial(W, q)
sage: KL.P(s2,s2*s1*s3*s2)
q + 1
R(x, y)

Return the Kazhdan-Lusztig \(R\) polynomial.

INPUT:

  • x, y – elements of the underlying Coxeter group

EXAMPLES:

sage: R.<q>=QQ[]
sage: W = WeylGroup("A2", prefix="s")
sage: [s1,s2]=W.simple_reflections()
sage: KL = KazhdanLusztigPolynomial(W, q)
sage: [KL.R(x,s2*s1) for x in [1,s1,s2,s1*s2]]
[q^2 - 2*q + 1, q - 1, q - 1, 0]
R_tilde(x, y)

Return the Kazhdan-Lusztig \(\tilde{R}\) polynomial.

Information about the \(\tilde{R}\) polynomials can be found in [Dy93] and [BB05].

INPUT:

  • x, y – elements of the underlying Coxeter group

EXAMPLES:

sage: R.<q> = QQ[]
sage: W = WeylGroup("A2", prefix="s")
sage: [s1,s2] = W.simple_reflections()
sage: KL = KazhdanLusztigPolynomial(W, q)
sage: [KL.R_tilde(x,s2*s1) for x in [1,s1,s2,s1*s2]]
[q^2, q, q, 0]

Previous topic

Hall Polynomials

Next topic

Posets

This Page