# Jack Symmetric Functions¶

Jack’s symmetric functions appear in [Ma1995] Chapter VI, section 10. Zonal polynomials are the subject of [Ma1995] Chapter VII. The parameter $$\alpha$$ in that reference is the parameter $$t$$ in this implementation in sage.

REFERENCES:

 [Jack1970] A class of symmetric functions with a parameter, Proc. R Soc. Edinburgh (A), 69, 1-18.
 [Ma1995] (1, 2) I. G. Macdonald, Symmetric functions and Hall polynomials, second ed., The Clarendon Press, Oxford University Press, New York, 1995, With contributions by A. Zelevinsky, Oxford Science Publications.
class sage.combinat.sf.jack.Jack(Sym, t='t')

The family of Jack symmetric functions including the $$P$$, $$Q$$, $$J$$, $$Qp$$ bases. The default parameter is t.

INPUT:

• self – the family of Jack symmetric function bases
• Sym – a ring of symmetric functions
• t – an optional parameter (default : ‘t’)

EXAMPLES:

sage: SymmetricFunctions(FractionField(QQ['t'])).jack()
Jack polynomials over Fraction Field of Univariate Polynomial Ring in t over Rational Field
sage: SymmetricFunctions(QQ).jack(1)
Jack polynomials with t=1 over Rational Field

J()

Returns the algebra of Jack polynomials in the $$J$$ basis.

INPUT:

• self – the family of Jack symmetric function bases

OUTPUT: the $$J$$ basis of the Jack symmetric functions

EXAMPLES:

sage: Sym = SymmetricFunctions(FractionField(QQ['t']))
sage: JJ = Sym.jack().J(); JJ
Symmetric Functions over Fraction Field of Univariate Polynomial Ring in t over Rational Field in the Jack J basis
sage: Sym = SymmetricFunctions(QQ)
sage: Sym.jack(t=-1).J()
Symmetric Functions over Rational Field in the Jack J with t=-1 basis


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: Sym = SymmetricFunctions(QQ)
sage: JJ = Sym.jack(t=1).J()
sage: s = Sym.schur()
sage: p = Partition([3,2,1,1])
sage: s(JJ(p)) == p.hook_product(1)*s(p)  # long time (4s on sage.math, 2012)
True


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

sage: Sym = SymmetricFunctions(QQ)
sage: JJ = Sym.jack(t=2).J()
sage: Z = Sym.zonal()
sage: p = Partition([2,2,1])
sage: Z(JJ(p)) == p.hook_product(2)*Z(p)
True

sage: Sym = SymmetricFunctions(FractionField(QQ['t']))
sage: JJ = Sym.jack().J()
sage: JP = Sym.jack().P()
sage: JJ(sum(JP(p) for p in Partitions(3)))
1/6*JackJ[1, 1, 1] + (1/(t+2))*JackJ[2, 1] + (1/(2*t^2+3*t+1))*JackJ[3]

sage: s = Sym.schur()
sage: JJ(s([3])) # indirect doctest
((t^2-3*t+2)/(6*t^2+18*t+12))*JackJ[1, 1, 1] + ((2*t-2)/(2*t^2+5*t+2))*JackJ[2, 1] + (1/(2*t^2+3*t+1))*JackJ[3]
sage: JJ(s([2,1]))
((t-1)/(3*t+6))*JackJ[1, 1, 1] + (1/(t+2))*JackJ[2, 1]
sage: JJ(s([1,1,1]))
1/6*JackJ[1, 1, 1]

P()

Returns the algebra of Jack polynomials in the $$P$$ basis.

INPUT:

• self – the family of Jack symmetric function bases

OUTPUT:

• the $$P$$ basis of the Jack symmetric functions

EXAMPLES:

sage: Sym = SymmetricFunctions(FractionField(QQ['t']))
sage: JP = Sym.jack().P(); JP
Symmetric Functions over Fraction Field of Univariate Polynomial Ring in t over Rational Field in the Jack P basis
sage: Sym.jack(t=-1).P()
Symmetric Functions over Fraction Field of Univariate Polynomial Ring in t over Rational Field in the Jack P with t=-1 basis


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

sage: Sym = SymmetricFunctions(QQ)
sage: JP = Sym.jack(t=1).P()
sage: s = Sym.schur()
sage: s(JP([2,2,1]))
s[2, 2, 1]
sage: JP(s([2,2,1]))
JackP[2, 2, 1]


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

sage: Sym = SymmetricFunctions(QQ)
sage: JP = Sym.jack(t=2).P()
sage: Z = Sym.zonal()
sage: Z(JP([2,2,1]))
Z[2, 2, 1]
sage: JP(Z[2, 2, 1])
JackP[2, 2, 1]
sage: JP([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: Sym = SymmetricFunctions(QQ['a','b'].fraction_field())
sage: (a,b) = Sym.base_ring().gens()
sage: Jacka = Sym.jack(t=a)
sage: Jackb = Sym.jack(t=b)
sage: m = Sym.monomial()
sage: JPa = Jacka.P()
sage: JPb = Jackb.P()
sage: m(JPa[2,1])
(6/(a+2))*m[1, 1, 1] + m[2, 1]
sage: m(JPb[2,1])
(6/(b+2))*m[1, 1, 1] + m[2, 1]
sage: m(a*JPb([2,1]) + b*JPa([2,1]))
((6*a^2+6*b^2+12*a+12*b)/(a*b+2*a+2*b+4))*m[1, 1, 1] + (a+b)*m[2, 1]
sage: JPa(JPb([2,1]))
((6*a-6*b)/(a*b+2*a+2*b+4))*JackP[1, 1, 1] + JackP[2, 1]

sage: Sym = SymmetricFunctions(FractionField(QQ['t']))
sage: JQ = Sym.jack().Q()
sage: JP = Sym.jack().P()
sage: JJ = Sym.jack().J()

sage: JP(JQ([2,1]))
((t+2)/(2*t^3+t^2))*JackP[2, 1]
sage: JP(JQ([3]))
((2*t^2+3*t+1)/(6*t^3))*JackP[3]
sage: JP(JQ([1,1,1]))
(6/(t^3+3*t^2+2*t))*JackP[1, 1, 1]

sage: JP(JJ([3]))
(2*t^2+3*t+1)*JackP[3]
sage: JP(JJ([2,1]))
(t+2)*JackP[2, 1]
sage: JP(JJ([1,1,1]))
6*JackP[1, 1, 1]

sage: s = Sym.schur()
sage: JP(s([2,1]))
((2*t-2)/(t+2))*JackP[1, 1, 1] + JackP[2, 1]
sage: s(_)
s[2, 1]

Q()

Returns the algebra of Jack polynomials in the $$Q$$ basis.

INPUT:

• self – the family of Jack symmetric function bases

OUTPUT:

• the $$Q$$ basis of the Jack symmetric functions

EXAMPLES:

sage: Sym = SymmetricFunctions(FractionField(QQ['t']))
sage: JQ = Sym.jack().Q(); JQ
Symmetric Functions over Fraction Field of Univariate Polynomial Ring in t over Rational Field in the Jack Q basis
sage: Sym = SymmetricFunctions(QQ)
sage: Sym.jack(t=-1).Q()
Symmetric Functions over Rational Field in the Jack Q with t=-1 basis

sage: Sym = SymmetricFunctions(FractionField(QQ['t']))
sage: JQ = Sym.jack().Q()
sage: JP = Sym.jack().P()
sage: JQ(sum(JP(p) for p in Partitions(3)))
(1/6*t^3+1/2*t^2+1/3*t)*JackQ[1, 1, 1] + ((2*t^3+t^2)/(t+2))*JackQ[2, 1] + (6*t^3/(2*t^2+3*t+1))*JackQ[3]

sage: s = Sym.schur()
sage: JQ(s([3])) # indirect doctest
(1/6*t^3-1/2*t^2+1/3*t)*JackQ[1, 1, 1] + ((2*t^3-2*t^2)/(t+2))*JackQ[2, 1] + (6*t^3/(2*t^2+3*t+1))*JackQ[3]
sage: JQ(s([2,1]))
(1/3*t^3-1/3*t)*JackQ[1, 1, 1] + ((2*t^3+t^2)/(t+2))*JackQ[2, 1]
sage: JQ(s([1,1,1]))
(1/6*t^3+1/2*t^2+1/3*t)*JackQ[1, 1, 1]

Qp()

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

INPUT:

• self – the family of Jack symmetric function bases

OUTPUT:

• the $$Q'$$ basis of the Jack symmetric functions

EXAMPLES:

sage: Sym = SymmetricFunctions(FractionField(QQ['t']))
sage: JP = Sym.jack().P()
sage: JQp = Sym.jack().Qp(); JQp
Symmetric Functions over Fraction Field of Univariate Polynomial Ring in t over Rational Field in the Jack Qp basis
sage: a = JQp([2])
sage: a.scalar(JP([2]))
1
sage: a.scalar(JP([1,1]))
0
sage: JP(JQp([2]))                        # todo: missing auto normalization
((t-1)/(t+1))*JackP[1, 1] + JackP[2]
sage: JP._normalize(JP(JQp([2])))
((t-1)/(t+1))*JackP[1, 1] + JackP[2]

base_ring()

Returns the base ring of the symmetric functions in which the Jack symmetric functions live

INPUT:

• self – the family of Jack symmetric function bases

OUTPUT:

• the base ring of the symmetric functions ring of self

EXAMPLES:

sage: J2 = SymmetricFunctions(QQ).jack(t=2)
sage: J2.base_ring()
Rational Field

symmetric_function_ring()

Returns the base ring of the symmetric functions of the Jack symmetric function bases

INPUT:

• self – the family of Jack symmetric function bases

OUTPUT:

• the symmetric functions ring of self

EXAMPLES:

sage: Jacks = SymmetricFunctions(FractionField(QQ['t'])).jack()
sage: Jacks.symmetric_function_ring()
Symmetric Functions over Fraction Field of Univariate Polynomial Ring in t over Rational Field

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.

This function is deprecated. Use instead: SymmetricFunctions(R).jack(t=value).J()

EXAMPLES

sage: JackPolynomialsJ(QQ)
doctest:1: DeprecationWarning: Deprecation warning: In the future use SymmetricFunctions(R).jack(t=t).J()
See http://trac.sagemath.org/5457 for details.
Symmetric Functions over Fraction Field of Univariate Polynomial Ring in t over Rational Field in the Jack J basis
sage: JackPolynomialsJ(QQ,t=-1)
doctest:1: DeprecationWarning: Deprecation warning: In the future use SymmetricFunctions(R).jack(t=-1).J()
See http://trac.sagemath.org/5457 for details.
Symmetric Functions over Rational Field in the Jack J with t=-1 basis


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); J
doctest:1: DeprecationWarning: Deprecation warning: In the future use SymmetricFunctions(R).jack(t=1).J()
See http://trac.sagemath.org/5457 for details.
Symmetric Functions over Rational Field in the Jack J with t=1 basis
sage: s = SymmetricFunctions(QQ).s()
sage: p = Partition([3,2,1,1])
sage: s(J(p)) == p.hook_product(1)*s(p)  # long time (4s on sage.math, 2012)
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)
doctest:1: DeprecationWarning: Deprecation warning: In the future use SymmetricFunctions(R).jack(t=2).J()
See http://trac.sagemath.org/5457 for details.
sage: Z = ZonalPolynomials(QQ)
doctest:1: DeprecationWarning: Deprecation warning: In the future use SymmetricFunctions(R).zonal()
See http://trac.sagemath.org/5457 for details.
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.

This function is deprecated. Use instead: SymmetricFunctions(R).jack(t=value).P()

EXAMPLES

sage: JackPolynomialsP(QQ)
doctest:1: DeprecationWarning: Deprecation warning: In the future use SymmetricFunctions(R).jack(t=t).P()
See http://trac.sagemath.org/5457 for details.
Symmetric Functions over Fraction Field of Univariate Polynomial Ring in t over Rational Field in the Jack P basis
sage: JackPolynomialsP(QQ,t=-1)
doctest:1: DeprecationWarning: Deprecation warning: In the future use SymmetricFunctions(R).jack(t=-1).P()
See http://trac.sagemath.org/5457 for details.
Symmetric Functions over Rational Field in the Jack P with t=-1 basis


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

sage: P = JackPolynomialsP(QQ,t=1); P
doctest:1: DeprecationWarning: Deprecation warning: In the future use SymmetricFunctions(R).jack(t=1).P()
See http://trac.sagemath.org/5457 for details.
Symmetric Functions over Rational Field in the Jack P with t=1 basis
sage: s = SymmetricFunctions(QQ).s()
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,t=2)
doctest:1: DeprecationWarning: Deprecation warning: In the future use SymmetricFunctions(R).jack(t=2).P()
See http://trac.sagemath.org/5457 for details.
sage: Z = ZonalPolynomials(QQ)
doctest:1: DeprecationWarning: Deprecation warning: In the future use SymmetricFunctions(R).zonal()
See http://trac.sagemath.org/5457 for details.
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.

This function is deprecated. Use instead: SymmetricFunctions(R).jack(t=value).Q()

EXAMPLES

sage: JackPolynomialsQ(QQ)
doctest:1: DeprecationWarning: Deprecation warning: In the future use SymmetricFunctions(R).jack(t=t).Q()
See http://trac.sagemath.org/5457 for details.
Symmetric Functions over Fraction Field of Univariate Polynomial Ring in t over Rational Field in the Jack Q basis
sage: JackPolynomialsQ(QQ,t=-1)
doctest:1: DeprecationWarning: Deprecation warning: In the future use SymmetricFunctions(R).jack(t=-1).Q()
See http://trac.sagemath.org/5457 for details.
Symmetric Functions over Rational Field in the Jack Q with t=-1 basis

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.

This function is deprecated. Use instead: SymmetricFunctions(R).jack(t=value).Qp()

EXAMPLES

sage: P = JackPolynomialsP(QQ)
doctest:1: DeprecationWarning: Deprecation warning: In the future use SymmetricFunctions(R).jack(t=t).P()
See http://trac.sagemath.org/5457 for details.
sage: Qp = JackPolynomialsQp(QQ)
doctest:1: DeprecationWarning: Deprecation warning: In the future use SymmetricFunctions(R).jack(t=t).Qp()
See http://trac.sagemath.org/5457 for details.
sage: a = Qp([2])
sage: a.scalar(P([2]))
1
sage: a.scalar(P([1,1]))
0
sage: s = P.realization_of().s()
sage: P(Qp([2]))                        # todo: missing auto normalization
((t-1)/(t+1))*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(jack)

A class of methods which are common to all Jack bases of the symmetric functions

INPUT:

• self – a Jack basis of the symmetric functions
• jack – a family of Jack symmetric function bases

EXAMPLES

sage: Sym = SymmetricFunctions(FractionField(QQ['t']))
sage: JP = Sym.jack().P(); JP.base_ring()
Fraction Field of Univariate Polynomial Ring in t over Rational Field
sage: Sym = SymmetricFunctions(QQ)
sage: JP = Sym.jack(t=2).P(); JP.base_ring()
Rational Field

class Element(M, x)

Create a combinatorial module element. This should never be called directly, but only through the parent combinatorial free module’s __call__() method.

TESTS:

sage: F = CombinatorialFreeModule(QQ, ['a','b','c'])
sage: B = F.basis()
sage: f = B['a'] + 3*B['c']; f
B['a'] + 3*B['c']
True

scalar_jack(x, t=None)

A scalar product where the power sums are orthogonal and $$\langle p_\mu, p_\mu \rangle = z_\mu t^{length(\mu)}$$

INPUT:

• self – an element of a Jack basis of the symmetric functions

• x – an element of the symmetric functions

• t – an optional parameter (default : None uses the parameter from

the basis)

OUTPUT:

• returns the Jack scalar product between x and self

EXAMPLES:

sage: Sym = SymmetricFunctions(FractionField(QQ['t']))
sage: JP = Sym.jack().P()
sage: JQ = Sym.jack().Q()
sage: p = Partitions(3).list()
sage: matrix([[JP(a).scalar_jack(JQ(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).

INPUT:

• self – a Jack basis of the symmetric functions
• part – a partition
• t – an optional parameter (default: uses the parameter $$t$$ from the Jack basis)

OUTPUT:

• a polynomial in the parameter t which is equal to the scalar product of J(part) and P(part)

EXAMPLES

sage: JP = SymmetricFunctions(FractionField(QQ['t'])).jack().P()
sage: JP.c1(Partition([2,1]))
t + 2

JackPolynomials_generic.c2(part)

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

INPUT:

• self – a Jack basis of the symmetric functions

• part – a partition

• t – an optional parameter (default: uses the parameter $$t$$ from the

Jack basis)

OUTPUT:

• a polynomial in the parameter t which is equal to the scalar product of J(part) and Q(part)

EXAMPLES:

sage: JP = SymmetricFunctions(FractionField(QQ['t'])).jack().P()
sage: JP.c2(Partition([2,1]))
2*t^3 + t^2

JackPolynomials_generic.coproduct_by_coercion(elt)

Returns the coproduct of the element elt by coercion to the Schur basis.

INPUT:

• self – a Jack symmetric function basis
• elt – an instance of this basis

OUTPUT:

• The coproduct acting on elt, the result is an element of the tensor squared of the Jack symmetric function basis

EXAMPLES:

sage: Sym = SymmetricFunctions(QQ['t'].fraction_field())
sage: Sym.jack().P()[2,2].coproduct() #indirect doctest
JackP[] # JackP[2, 2] + (2/(t+1))*JackP[1] # JackP[2, 1] + ((8*t+4)/(t^3+4*t^2+5*t+2))*JackP[1, 1] # JackP[1, 1] + JackP[2] # JackP[2] + (2/(t+1))*JackP[2, 1] # JackP[1] + JackP[2, 2] # JackP[]

JackPolynomials_generic.jack_family()

Returns the family of Jack bases associated to the basis self

INPUT:

• self – a Jack basis of the symmetric functions

OUTPUT:

• the family of Jack symmetric functions associated to self

EXAMPLES:

sage: JackP = SymmetricFunctions(QQ).jack(t=2).P()
sage: JackP.jack_family()
Jack polynomials with t=2 over Rational Field

class sage.combinat.sf.jack.JackPolynomials_j(jack)

The $$J$$ basis is a defined as a normalized form of the $$P$$ basis

INPUT:

• self – an instance of the Jack $$P$$ basis of the symmetric functions
• jack – a family of Jack symmetric function bases

EXAMPLES:

sage: J = SymmetricFunctions(FractionField(QQ['t'])).jack().J()
sage: TestSuite(J).run(skip=['_test_associativity', '_test_distributivity', '_test_prod']) # products are too expensive
sage: TestSuite(J).run(elements = [J.t*J[1,1]+J[2], J[1]+(1+J.t)*J[1,1]])  # long time (3s on sage.math, 2012)

class Element(M, x)

Create a combinatorial module element. This should never be called directly, but only through the parent combinatorial free module’s __call__() method.

TESTS:

sage: F = CombinatorialFreeModule(QQ, ['a','b','c'])
sage: B = F.basis()
sage: f = B['a'] + 3*B['c']; f
B['a'] + 3*B['c']
True

class sage.combinat.sf.jack.JackPolynomials_p(jack)

The $$P$$ basis is uni-triangularly related to the monomial basis and orthogonal with respect to the Jack scalar product.

INPUT:

• self – an instance of the Jack $$P$$ basis of the symmetric functions
• jack – a family of Jack symmetric function bases

EXAMPLES:

sage: P = SymmetricFunctions(FractionField(QQ['t'])).jack().P()
sage: TestSuite(P).run(skip=['_test_associativity', '_test_distributivity', '_test_prod']) # products are too expensive
sage: TestSuite(P).run(elements = [P.t*P[1,1]+P[2], P[1]+(1+P.t)*P[1,1]])

class Element(M, x)

Create a combinatorial module element. This should never be called directly, but only through the parent combinatorial free module’s __call__() method.

TESTS:

sage: F = CombinatorialFreeModule(QQ, ['a','b','c'])
sage: B = F.basis()
sage: f = B['a'] + 3*B['c']; f
B['a'] + 3*B['c']
True

scalar_jack(x, t=None)

The scalar product on the symmetric functions where the power sums are orthogonal and $$\langle p_\mu, p_\mu \rangle = z_\mu t^{length(mu)}$$ where the t parameter from the Jack symmetric function family.

INPUT:

• self – an element of the Jack $$P$$ basis
• x – an element of the $$P$$ basis

EXAMPLES

sage: JP = SymmetricFunctions(FractionField(QQ['t'])).jack().P()
sage: l = [JP(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 $$P(part2)$$.

This is equation (10.16) of [Mc1995] on page 380.

INPUT:

• self – an instance of the Jack $$P$$ basis of the symmetric functions
• part1 – a partition
• part2 – an optional partition (default : None)

OUTPUT:

• the scalar product between $$P(part1)$$ and $$P(part2)$$ (or itself if $$part2$$ is None)

REFRENCES:

 [Mc1995] I. G. Macdonald, Symmetric functions and Hall polynomials, second ed., The Clarendon Press, Oxford University Press, New York, 1995, With contributions by A. Zelevinsky, Oxford Science Publications.

EXAMPLES:

sage: JP = SymmetricFunctions(FractionField(QQ['t'])).jack().P()
sage: JJ = SymmetricFunctions(FractionField(QQ['t'])).jack().J()
sage: JP.scalar_jack_basis(Partition([2,1]), Partition([1,1,1]))
0
sage: JP._normalize_coefficients(JP.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)
sage: JJ(JP[3,2,1]).scalar_jack(JP[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: JP.scalar_jack_basis(Partition([2,1]), Partition([2,1]))
(2*t^3 + t^2)/(t + 2)
sage: JJ(JP[2,1]).scalar_jack(JP[2,1])
(2*t^3 + t^2)/(t + 2)

class sage.combinat.sf.jack.JackPolynomials_q(jack)

The $$Q$$ basis is defined as a normalized form of the $$P$$ basis

INPUT:

• self – an instance of the Jack $$Q$$ basis of the symmetric functions
• jack – a family of Jack symmetric function bases

EXAMPLES:

sage: Q = SymmetricFunctions(FractionField(QQ['t'])).jack().Q()
sage: TestSuite(Q).run(skip=['_test_associativity', '_test_distributivity', '_test_prod']) # products are too expensive
sage: TestSuite(Q).run(elements = [Q.t*Q[1,1]+Q[2], Q[1]+(1+Q.t)*Q[1,1]])  # long time (3s on sage.math, 2012)

class Element(M, x)

Create a combinatorial module element. This should never be called directly, but only through the parent combinatorial free module’s __call__() method.

TESTS:

sage: F = CombinatorialFreeModule(QQ, ['a','b','c'])
sage: B = F.basis()
sage: f = B['a'] + 3*B['c']; f
B['a'] + 3*B['c']
True

class sage.combinat.sf.jack.JackPolynomials_qp(jack)

The $$Qp$$ basis is the dual basis to the $$P$$ basis with respect to the standard scalar product

INPUT:

• self – an instance of the Jack $$Qp$$ basis of the symmetric functions
• jack – a family of Jack symmetric function bases

EXAMPLES:

sage: Qp = SymmetricFunctions(FractionField(QQ['t'])).jack().Qp()
sage: TestSuite(Qp).run(skip=['_test_associativity', '_test_distributivity', '_test_prod']) # products are too expensive
sage: TestSuite(Qp).run(elements = [Qp.t*Qp[1,1]+Qp[2], Qp[1]+(1+Qp.t)*Qp[1,1]])  # long time (3s on sage.math, 2012)

class Element(M, x)

Create a combinatorial module element. This should never be called directly, but only through the parent combinatorial free module’s __call__() method.

TESTS:

sage: F = CombinatorialFreeModule(QQ, ['a','b','c'])
sage: B = F.basis()
sage: f = B['a'] + 3*B['c']; f
B['a'] + 3*B['c']
True

JackPolynomials_qp.coproduct_by_coercion(elt)

Returns the coproduct of the element elt by coercion to the Schur basis.

INPUT:

• elt – an instance of the Qp basis

OUTPUT:

• The coproduct acting on elt, the result is an element of the tensor squared of the Qp symmetric function basis

EXAMPLES:

sage: Sym = SymmetricFunctions(QQ['t'].fraction_field())
sage: JQp = Sym.jack().Qp()
sage: JQp[2,2].coproduct()   #indirect doctest
JackQp[] # JackQp[2, 2] + (2*t/(t+1))*JackQp[1] # JackQp[2, 1] + JackQp[1, 1] # JackQp[1, 1] + ((4*t^3+8*t^2)/(2*t^3+5*t^2+4*t+1))*JackQp[2] # JackQp[2] + (2*t/(t+1))*JackQp[2, 1] # JackQp[1] + JackQp[2, 2] # JackQp[]

sage.combinat.sf.jack.NoneConvention(R, t)

Helper function to mimic behavior of old conventions.

INPUT:

• R – ring
• t – parameter

EXAMPLES

sage: sage.combinat.sf.jack.NoneConvention(QQ, None)
(Fraction Field of Univariate Polynomial Ring in t over Rational Field, t)
sage: R = QQ['t']
sage: sage.combinat.sf.jack.NoneConvention(R, R.gen())
(Univariate Polynomial Ring in t over Rational Field, t)

class sage.combinat.sf.jack.SymmetricFunctionAlgebra_zonal(Sym)

Returns the algebra of zonal polynomials.

INPUT:

• self – a zonal basis of the symmetric functions
• Sym – a ring of the symmetric functions

EXAMPLES

sage: Z = SymmetricFunctions(QQ).zonal()
sage: Z([2])^2
64/45*Z[2, 2] + 16/21*Z[3, 1] + Z[4]
sage: Z = SymmetricFunctions(QQ).zonal()
sage: TestSuite(Z).run(skip=['_test_associativity', '_test_distributivity', '_test_prod']) # products are too expensive
sage: TestSuite(Z).run(elements = [Z[1,1]+Z[2], Z[1]+2*Z[1,1]])

class Element(M, x)

Create a combinatorial module element. This should never be called directly, but only through the parent combinatorial free module’s __call__() method.

TESTS:

sage: F = CombinatorialFreeModule(QQ, ['a','b','c'])
sage: B = F.basis()
sage: f = B['a'] + 3*B['c']; f
B['a'] + 3*B['c']
True

scalar_zonal(x)

The zonal scalar product has the power sum basis and the zonal symmetric functions are orthogonal. In particular, $$\langle p_\mu, p_\mu \rangle = z_\mu 2^{length(\mu)}$$.

INPUT:

• self – an element of the zonal basis
• x – an element of the symmetric function

OUTPUT:

• the scalar product between self and x

EXAMPLES

sage: Sym = SymmetricFunctions(QQ)
sage: Z = Sym.zonal()
sage: parts = Partitions(3).list()
sage: matrix([[Z(a).scalar_zonal(Z(b)) for a in parts] for b in parts])
[16/5    0    0]
[   0    5    0]
[   0    0    4]
sage: p = Z.symmetric_function_ring().power()
sage: matrix([[Z(p(a)).scalar_zonal(p(b)) for a in parts] for b in parts])
[ 6  0  0]
[ 0  8  0]
[ 0  0 48]

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

Returns the algebra of zonal polynomials.

This function is deprecated. Use instead: SymmetricFunctions(R).zonal()

EXAMPLES

sage: Z = ZonalPolynomials(QQ)
doctest:1: DeprecationWarning: Deprecation warning: In the future use SymmetricFunctions(R).zonal()
See http://trac.sagemath.org/5457 for details.
sage: a = Z([2])
sage: Z([2])^2
64/45*Z[2, 2] + 16/21*Z[3, 1] + Z[4]

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

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

INPUT:

• part – a partition
• t – an optional parameter (default: uses the parameter $$t$$ from the Jack basis)

OUTPUT:

• a polynomial in the parameter t which is equal to the scalar product of J(part) and P(part)

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)

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

INPUT:

• self – a Jack basis of the symmetric functions
• part – a partition
• t – an optional parameter (default: uses the parameter $$t$$ from the Jack basis)

OUTPUT:

• a polynomial in the parameter t which is equal to the scalar product of J(part) and Q(part)

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.normalize_coefficients(self, c)

If our coefficient ring is the field of fractions over a univariate polynomial ring over the rationals, then we should clear both the numerator and denominator of the denominators of their coefficients.

INPUT:

• self – a Jack basis of the symmetric functions
• c – a coefficient in the base ring of self

OUTPUT:

• divide numerator and denominator by the greatest common divisor

EXAMPLES:

sage: JP = SymmetricFunctions(FractionField(QQ['t'])).jack().P()
sage: t = JP.base_ring().gen()
sage: a = 2/(1/2*t+1/2)
sage: JP._normalize_coefficients(a)
4/(t + 1)
sage: a = 1/(1/3+1/6*t)
sage: JP._normalize_coefficients(a)
6/(t + 2)
sage: a = 24/(4*t^2 + 12*t + 8)
sage: JP._normalize_coefficients(a)
6/(t^2 + 3*t + 2)

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

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

INPUT:

• part1, part2 – two partitions
• t – a parameter

OUTPUT:

• returns the scalar product between the power sum indexed by part1 and part2

EXAMPLES:

sage: Q.<t> = QQ[]
sage: from sage.combinat.sf.jack import part_scalar_jack
sage: matrix([[part_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]


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