Kazhdan-Lusztig Polynomials

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.


  • W – a Weyl Group
  • q – an indeterminate


  • 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.


[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


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)
q + 1

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
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.


  • x, y – elements of the underlying Coxeter group

See also

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


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.


  • x, y – elements of the underlying Coxeter group


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]
sage.combinat.kazhdan_lusztig.laurent_polynomial_truncate(p, n)

Truncate the Laurent polynomial p, returning only terms of degree less than n, similar to the truncate method for polynomials.


sage: from sage.combinat.kazhdan_lusztig import laurent_polynomial_truncate
sage: P.<q> = LaurentPolynomialRing(QQ)
sage: laurent_polynomial_truncate((q+q^-1)^3+q^2*(q+q^-1)^4,3)
6*q^2 + 3*q + 4 + 3*q^-1 + q^-2 + q^-3

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