Jack Polynomials

sage.combinat.sf.jack.JackPolynomialsJ(R, t=None)

Returns the algebra of Jack polynomials in the J basis.

If t is not specified, then the base ring will be obtained by making the univariate polynomial ring over R with the variable t and taking its fraction field.

EXAMPLES:

sage: JackPolynomialsJ(QQ)
Jack polynomials in the J basis over Fraction Field of Univariate Polynomial Ring in t over Rational Field
sage: JackPolynomialsJ(QQ,t=-1)
Jack polynomials in the J basis with t=-1 over Rational Field

At t = 1, the Jack polynomials in the J basis are scalar multiples of the Schur functions with the scalar given by a Partition’s hook_product method at 1.

sage: J = JackPolynomialsJ(QQ, t=1)
sage: s = SFASchur(J.base_ring())
sage: p = Partition([3,2,1,1])
sage: s(J(p)) == p.hook_product(1)*s(p)
True

At t = 2, the Jack polynomials on the J basis are scalar multiples of the zonal polynomials with the scalar given by a Partition’s hook_product method at 1.

sage: t = 2
sage: J = JackPolynomialsJ(QQ,t=t)
sage: Z = ZonalPolynomials(J.base_ring())
sage: p = Partition([2,2,1])
sage: Z(J(p)) == p.hook_product(t)*Z(p)
True

sage.combinat.sf.jack.JackPolynomialsP(R, t=None)

Returns the algebra of Jack polynomials in the P basis.

If t is not specified, then the base ring will be obtained by making the univariate polynomial ring over R with the variable t and taking its fraction field.

EXAMPLES:

sage: JackPolynomialsP(QQ)
Jack polynomials in the P basis over Fraction Field of Univariate Polynomial Ring in t over Rational Field
sage: JackPolynomialsP(QQ,t=-1)
Jack polynomials in the P basis with t=-1 over Rational Field

At t = 1, the Jack polynomials on the P basis are the Schur symmetric functions.

sage: P = JackPolynomialsP(QQ,1)
sage: s = SFASchur(QQ)
sage: P([2,1])^2
JackP[2, 2, 1, 1] + JackP[2, 2, 2] + JackP[3, 1, 1, 1] + 2*JackP[3, 2, 1] + JackP[3, 3] + JackP[4, 1, 1] + JackP[4, 2]
sage: s([2,1])^2
s[2, 2, 1, 1] + s[2, 2, 2] + s[3, 1, 1, 1] + 2*s[3, 2, 1] + s[3, 3] + s[4, 1, 1] + s[4, 2]

At t = 2, the Jack polynomials on the P basis are the zonal polynomials.

sage: P = JackPolynomialsP(QQ,2)
sage: Z = ZonalPolynomials(QQ)
sage: P([2])^2
64/45*JackP[2, 2] + 16/21*JackP[3, 1] + JackP[4]
sage: Z([2])^2
64/45*Z[2, 2] + 16/21*Z[3, 1] + Z[4]
sage: Z(P([2,1]))
Z[2, 1]
sage: P(Z([2,1]))
JackP[2, 1]

sage.combinat.sf.jack.JackPolynomialsQ(R, t=None)

Returns the algebra of Jack polynomials in the Q basis.

If t is not specified, then the base ring will be obtained by making the univariate polynomial ring over R with the variable t and taking its fraction field.

EXAMPLES:

sage: JackPolynomialsQ(QQ)
Jack polynomials in the Q basis over Fraction Field of Univariate Polynomial Ring in t over Rational Field
sage: JackPolynomialsQ(QQ,t=-1)
Jack polynomials in the Q basis with t=-1 over Rational Field

sage.combinat.sf.jack.JackPolynomialsQp(R, t=None)

Returns the algebra of Jack polynomials in the Qp, which is dual to the P basis with respect to the standard scalar product.

If t is not specified, then the base ring will be obtained by making the univariate polynomial ring over R with the variable t and taking its fraction field.

EXAMPLES:

sage: P = JackPolynomialsP(QQ)
sage: Qp = JackPolynomialsQp(QQ)
sage: a = Qp([2])
sage: a.scalar(P([2]))
1
sage: a.scalar(P([1,1]))
0
sage: P(Qp([2]))                        # todo: missing auto normalization
((2*t-2)/(2*t+2))*JackP[1, 1] + JackP[2]
sage: P._normalize(P(Qp([2])))
((t-1)/(t+1))*JackP[1, 1] + JackP[2]

class sage.combinat.sf.jack.JackPolynomials_generic(R, t=None)

Bases: sage.combinat.sf.sfa.SymmetricFunctionAlgebra_generic

class Element(M, x)

Bases: sage.combinat.sf.sfa.SymmetricFunctionAlgebra_generic_Element

scalar_jack(x)

EXAMPLES:

sage: P = JackPolynomialsP(QQ)
sage: Q = JackPolynomialsQ(QQ)
sage: p = Partitions(3).list()
sage: matrix([[P(a).scalar_jack(Q(b)) for a in p] for b in p])
[1 0 0]
[0 1 0]
[0 0 1]

JackPolynomials_generic.c1(part)

Returns the t-Jack scalar product between J(part) and P(part).

EXAMPLES:

sage: P = JackPolynomialsP(QQ)
sage: P.c1(Partition([2,1]))
t + 2

JackPolynomials_generic.c2(part)

Returns the t-Jack scalar product between J(part) and Q(part).

EXAMPLES:

sage: P = JackPolynomialsP(QQ)
sage: P.c2(Partition([2,1]))
2*t^3 + t^2

class sage.combinat.sf.jack.JackPolynomials_j(R, t=None)

Bases: sage.combinat.sf.jack.JackPolynomials_generic

class Element(M, x)

Bases: sage.combinat.sf.jack.JackPolynomials_generic.Element

class sage.combinat.sf.jack.JackPolynomials_p(R, t=None)

Bases: sage.combinat.sf.jack.JackPolynomials_generic

class Element(M, x)

Bases: sage.combinat.sf.jack.JackPolynomials_generic.Element

scalar_jack(x)

EXAMPLES:

sage: P = JackPolynomialsP(QQ) sage: l = [P(p) for p in Partitions(3)] sage: matrix([[a.scalar_jack(b) for a in l] for b in l]) [ 6*t^3/(2*t^2 + 3*t + 1) 0 0] [ 0 (2*t^3 + t^2)/(t + 2) 0] [ 0 0 1/6*t^3 + 1/2*t^2 + 1/3*t]

JackPolynomials_p.scalar_jack_basis(part1, part2=None)

Returns the scalar product of P[part1] and P[part2].

Todo: check all the results!

EXAMPLES:

sage: P = JackPolynomialsP(QQ) sage: P.scalar_jack_basis(Partition([2,1]), Partition([1,1,1])) 0 sage: P._normalize_coefficients(P.scalar_jack_basis(Partition([3,2,1]), Partition([3,2,1]))) (12*t^6 + 20*t^5 + 11*t^4 + 2*t^3)/(2*t^3 + 11*t^2 + 20*t + 12)

With a single argument, takes part2 = part1

sage: P.scalar_jack_basis(Partition([2,1]), Partition([2,1])) (2*t^3 + t^2)/(t + 2)

NT: those results do not quite with Macdonald Symmetric Function and Orthogonal Polynomials p.12 (11.3). Is this P basis a normalization variant of that of Macdo?

class sage.combinat.sf.jack.JackPolynomials_q(R, t=None)

Bases: sage.combinat.sf.jack.JackPolynomials_generic

class Element(M, x)

Bases: sage.combinat.sf.jack.JackPolynomials_generic.Element

class sage.combinat.sf.jack.ZonalPolynomials(R)

Bases: sage.combinat.sf.jack.JackPolynomials_p

class Element(M, x)

Bases: sage.combinat.sf.jack.JackPolynomials_p.Element

sage.combinat.sf.jack.c1(part, t)

EXAMPLES:

sage: from sage.combinat.sf.jack import c1
sage: t = QQ['t'].gen()
sage: [c1(p,t) for p in Partitions(3)]
[2*t^2 + 3*t + 1, t + 2, 6]

sage.combinat.sf.jack.c2(part, t)

EXAMPLES:

sage: from sage.combinat.sf.jack import c2
sage: t = QQ['t'].gen()
sage: [c2(p,t) for p in Partitions(3)]
[6*t^3, 2*t^3 + t^2, t^3 + 3*t^2 + 2*t]

sage.combinat.sf.jack.scalar_jack(part1, part2, t)

Returns the Jack scalar product between p(part1) and p(part2) where p is the power-sum basis.

EXAMPLES:

sage: Q.<t> = QQ[]
sage: from sage.combinat.sf.jack import scalar_jack
sage: matrix([[scalar_jack(p1,p2,t) for p1 in Partitions(4)] for p2 in Partitions(4)])
[   4*t      0      0      0      0]
[     0  3*t^2      0      0      0]
[     0      0  8*t^2      0      0]
[     0      0      0  4*t^3      0]
[     0      0      0      0 24*t^4]