Kazhdan-Lusztig Polynomials

class sage.combinat.kazhdan_lusztig.KazhdanLusztigPolynomial(W, q, trace=False)

Bases: sage.structure.sage_object.SageObject

P(*args, **kwds)

Returns 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.

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(*args, **kwds)

Returns the Kazhdan-Lusztig R polynomial.

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]

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

Truncates the Laurent polynomial p, returning only terms of degree <n, similar to the truncate method for polynomials.

EXAMPLES

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