Macdonald Polynomials#

Notation used in the definitions follows mainly [Mac1995].

The integral forms of the bases \(H\) and \(Ht\) do not appear in Macdonald’s book. They correspond to the two bases \(H_\mu[X;q,t] = \sum_{\nu} K_{\nu\mu}(q,t) s_\mu[X]\) and \({\tilde H}_\mu[X;q,t] = t^{n(\mu)} \sum_{\nu} K_{\nu\mu}(q,1/t) s_\nu[X]\) where \(K_{\mu\nu}(q,t)\) are the Macdonald \(q,t\)-Koskta coefficients.

The \(Ht\) in this case is short for \({\tilde H}\) and is the basis which is the graded Frobenius image of the Garsia-Haiman modules [GH1993].

REFERENCES:

[Mac1995]

[GH1993]

A. Garsia, M. Haiman, A graded representation module for Macdonald’s polynomials, Proc. Nat. Acad. U.S.A. no. 90, 3607–3610.

[BGHT1999] (1,2)

F. Bergeron, A. M. Garsia, M. Haiman, and G. Tesler, Identities and positivity conjectures for some remarkable operators in the theory of symmetric functions, Methods Appl. Anal. 6 (1999), no. 3, 363–420.

[LLM1998] (1,2)

L. Lapointe, A. Lascoux, J. Morse, Determinantal Expressions for Macdonald Polynomials, IRMN no. 18 (1998). arXiv math/9808050.

[BH2013] (1,2)

F. Bergeron, M. Haiman, Tableaux Formulas for Macdonald Polynomials, Special edition in honor of Christophe Reutenauer 60 birthday, International Journal of Algebra and Computation, Volume 23, Issue 4, (2013), pp. 833-852.

class sage.combinat.sf.macdonald.Macdonald(Sym, q, t)[source]#

Bases: UniqueRepresentation

Macdonald Symmetric functions including \(P\), \(Q\), \(J\), \(H\), \(Ht\) bases also including the S basis which is the plethystic transformation of the Schur basis (that which is dual to the Schur basis with respect to the Macdonald \(q,t\)-scalar product)

INPUT:

self – a family of Macdonald symmetric function bases

EXAMPLES:

Sage

sage: t = QQ['t'].gen(); SymmetricFunctions(QQ['t'].fraction_field()).macdonald(q=t,t=1)
Macdonald polynomials with q=t and t=1 over Fraction Field of Univariate Polynomial Ring in t over Rational Field
sage: Sym = SymmetricFunctions(FractionField(QQ['t'])).macdonald()
Traceback (most recent call last):
...
TypeError: unable to evaluate 'q' in Fraction Field of Univariate Polynomial Ring in t over Rational Field

Python

>>> from sage.all import *
>>> t = QQ['t'].gen(); SymmetricFunctions(QQ['t'].fraction_field()).macdonald(q=t,t=Integer(1))
Macdonald polynomials with q=t and t=1 over Fraction Field of Univariate Polynomial Ring in t over Rational Field
>>> Sym = SymmetricFunctions(FractionField(QQ['t'])).macdonald()
Traceback (most recent call last):
...
TypeError: unable to evaluate 'q' in Fraction Field of Univariate Polynomial Ring in t over Rational Field

H()[source]#

Returns the Macdonald polynomials on the H basis. When the \(H\) basis is expanded on the Schur basis, the coefficients are the \(qt\)-Kostka numbers.

INPUT:

self – a family of Macdonald symmetric function bases

OUTPUT:

returns the \(H\) Macdonald basis of symmetric functions

EXAMPLES:

Sage

sage: Sym = SymmetricFunctions(FractionField(QQ['q','t']))
sage: H = Sym.macdonald().H(); H
Symmetric Functions over Fraction Field of Multivariate Polynomial Ring in q, t over Rational Field in the Macdonald H basis
sage: s = Sym.schur()
sage: s(H([2]))
q*s[1, 1] + s[2]
sage: s(H([1,1]))
s[1, 1] + t*s[2]

Python

>>> from sage.all import *
>>> Sym = SymmetricFunctions(FractionField(QQ['q','t']))
>>> H = Sym.macdonald().H(); H
Symmetric Functions over Fraction Field of Multivariate Polynomial Ring in q, t over Rational Field in the Macdonald H basis
>>> s = Sym.schur()
>>> s(H([Integer(2)]))
q*s[1, 1] + s[2]
>>> s(H([Integer(1),Integer(1)]))
s[1, 1] + t*s[2]

Coercions to/from the Schur basis are implemented:

Sage

sage: H = Sym.macdonald().H()
sage: s = Sym.schur()
sage: H(s([2]))
(q/(q*t-1))*McdH[1, 1] - (1/(q*t-1))*McdH[2]

Python

>>> from sage.all import *
>>> H = Sym.macdonald().H()
>>> s = Sym.schur()
>>> H(s([Integer(2)]))
(q/(q*t-1))*McdH[1, 1] - (1/(q*t-1))*McdH[2]

Ht()[source]#

Returns the Macdonald polynomials on the \(Ht\) basis. The elements of the \(Ht\) basis are eigenvectors of the \(nabla\) operator. When expanded on the Schur basis, the coefficients are the modified \(qt\)-Kostka numbers.

INPUT:

self – a family of Macdonald symmetric function bases

OUTPUT:

returns the \(Ht\) Macdonald basis of symmetric functions

EXAMPLES:

Sage

sage: Sym = SymmetricFunctions(FractionField(QQ['q','t']))
sage: Ht = Sym.macdonald().Ht(); Ht
Symmetric Functions over Fraction Field of Multivariate Polynomial Ring in q, t over Rational Field in the Macdonald Ht basis
sage: [Ht(p).nabla() for p in Partitions(3)]
[q^3*McdHt[3], q*t*McdHt[2, 1], t^3*McdHt[1, 1, 1]]

Python

>>> from sage.all import *
>>> Sym = SymmetricFunctions(FractionField(QQ['q','t']))
>>> Ht = Sym.macdonald().Ht(); Ht
Symmetric Functions over Fraction Field of Multivariate Polynomial Ring in q, t over Rational Field in the Macdonald Ht basis
>>> [Ht(p).nabla() for p in Partitions(Integer(3))]
[q^3*McdHt[3], q*t*McdHt[2, 1], t^3*McdHt[1, 1, 1]]

Sage

sage: s = Sym.schur()
sage: from sage.combinat.sf.macdonald import qt_kostka
sage: q,t = Ht.base_ring().gens()
sage: s(Ht([2,1]))
q*t*s[1, 1, 1] + (q+t)*s[2, 1] + s[3]
sage: qt_kostka([1,1,1],[2,1]).subs(t=1/t)*t^Partition([2,1]).weighted_size()
q*t
sage: qt_kostka([2,1],[2,1]).subs(t=1/t)*t^Partition([2,1]).weighted_size()
q + t
sage: qt_kostka([3],[2,1]).subs(t=1/t)*t^Partition([2,1]).weighted_size()
1

Python

>>> from sage.all import *
>>> s = Sym.schur()
>>> from sage.combinat.sf.macdonald import qt_kostka
>>> q,t = Ht.base_ring().gens()
>>> s(Ht([Integer(2),Integer(1)]))
q*t*s[1, 1, 1] + (q+t)*s[2, 1] + s[3]
>>> qt_kostka([Integer(1),Integer(1),Integer(1)],[Integer(2),Integer(1)]).subs(t=Integer(1)/t)*t**Partition([Integer(2),Integer(1)]).weighted_size()
q*t
>>> qt_kostka([Integer(2),Integer(1)],[Integer(2),Integer(1)]).subs(t=Integer(1)/t)*t**Partition([Integer(2),Integer(1)]).weighted_size()
q + t
>>> qt_kostka([Integer(3)],[Integer(2),Integer(1)]).subs(t=Integer(1)/t)*t**Partition([Integer(2),Integer(1)]).weighted_size()
1

Coercions to/from the Schur basis are implemented:

Sage

sage: Ht = Sym.macdonald().Ht()
sage: s = Sym.schur()
sage: Ht(s([2,1]))
(q/(q*t^2-t^3-q^2+q*t))*McdHt[1, 1, 1] + ((-q^2-q*t-t^2)/(q^2*t^2-q^3-t^3+q*t))*McdHt[2, 1] + (t/(-q^3+q^2*t+q*t-t^2))*McdHt[3]
sage: Ht(s([2]))
((-q)/(-q+t))*McdHt[1, 1] + (t/(-q+t))*McdHt[2]

Python

>>> from sage.all import *
>>> Ht = Sym.macdonald().Ht()
>>> s = Sym.schur()
>>> Ht(s([Integer(2),Integer(1)]))
(q/(q*t^2-t^3-q^2+q*t))*McdHt[1, 1, 1] + ((-q^2-q*t-t^2)/(q^2*t^2-q^3-t^3+q*t))*McdHt[2, 1] + (t/(-q^3+q^2*t+q*t-t^2))*McdHt[3]
>>> Ht(s([Integer(2)]))
((-q)/(-q+t))*McdHt[1, 1] + (t/(-q+t))*McdHt[2]

J()[source]#

Returns the Macdonald polynomials on the \(J\) basis also known as the integral form of the Macdonald polynomials. These are scalar multiples of both the \(P\) and \(Q\) bases. When expressed in the \(P\) or \(Q\) basis, the scaling coefficients are polynomials in \(q\) and \(t\) rather than rational functions.

The \(J\) basis is calculated using determinantal formulas of Lapointe-Lascoux-Morse giving the action on the S-basis [LLM1998].

INPUT:

self – a family of Macdonald symmetric function bases

OUTPUT:

returns the \(J\) Macdonald basis of symmetric functions

EXAMPLES:

Sage

sage: Sym = SymmetricFunctions(FractionField(QQ['q','t']))
sage: J = Sym.macdonald().J(); J
Symmetric Functions over Fraction Field of Multivariate Polynomial Ring in q, t over Rational Field in the Macdonald J basis
sage: P = Sym.macdonald().P()
sage: Q = Sym.macdonald().Q()
sage: P(J([2]))
(q*t^2-q*t-t+1)*McdP[2]
sage: P(J([1,1]))
(t^3-t^2-t+1)*McdP[1, 1]
sage: Q(J([2]))
(q^3-q^2-q+1)*McdQ[2]
sage: Q(J([1,1]))
(q^2*t-q*t-q+1)*McdQ[1, 1]

Python

>>> from sage.all import *
>>> Sym = SymmetricFunctions(FractionField(QQ['q','t']))
>>> J = Sym.macdonald().J(); J
Symmetric Functions over Fraction Field of Multivariate Polynomial Ring in q, t over Rational Field in the Macdonald J basis
>>> P = Sym.macdonald().P()
>>> Q = Sym.macdonald().Q()
>>> P(J([Integer(2)]))
(q*t^2-q*t-t+1)*McdP[2]
>>> P(J([Integer(1),Integer(1)]))
(t^3-t^2-t+1)*McdP[1, 1]
>>> Q(J([Integer(2)]))
(q^3-q^2-q+1)*McdQ[2]
>>> Q(J([Integer(1),Integer(1)]))
(q^2*t-q*t-q+1)*McdQ[1, 1]

Coercions from the \(Q\) and \(J\) basis (proportional) and to/from the Schur basis are implemented:

Sage

sage: P = Sym.macdonald().P()
sage: Q = Sym.macdonald().Q()
sage: J = Sym.macdonald().J()
sage: s = Sym.schur()

Python

>>> from sage.all import *
>>> P = Sym.macdonald().P()
>>> Q = Sym.macdonald().Q()
>>> J = Sym.macdonald().J()
>>> s = Sym.schur()

Sage

sage: J(P([2]))
(1/(q*t^2-q*t-t+1))*McdJ[2]

Python

>>> from sage.all import *
>>> J(P([Integer(2)]))
(1/(q*t^2-q*t-t+1))*McdJ[2]

Sage

sage: J(Q([2]))
(1/(q^3-q^2-q+1))*McdJ[2]

Python

>>> from sage.all import *
>>> J(Q([Integer(2)]))
(1/(q^3-q^2-q+1))*McdJ[2]

Sage

sage: s(J([2]))
(-q*t+t^2+q-t)*s[1, 1] + (q*t^2-q*t-t+1)*s[2]
sage: J(s([2]))
((q-t)/(q*t^4-q*t^3-q*t^2-t^3+q*t+t^2+t-1))*McdJ[1, 1] + (1/(q*t^2-q*t-t+1))*McdJ[2]

Python

>>> from sage.all import *
>>> s(J([Integer(2)]))
(-q*t+t^2+q-t)*s[1, 1] + (q*t^2-q*t-t+1)*s[2]
>>> J(s([Integer(2)]))
((q-t)/(q*t^4-q*t^3-q*t^2-t^3+q*t+t^2+t-1))*McdJ[1, 1] + (1/(q*t^2-q*t-t+1))*McdJ[2]

P()[source]#

Returns Macdonald polynomials in \(P\) basis. The \(P\) basis is defined here as a normalized form of the \(J\) basis.

INPUT:

self – a family of Macdonald symmetric function bases

OUTPUT:

returns the \(P\) Macdonald basis of symmetric functions

EXAMPLES:

Sage

sage: Sym = SymmetricFunctions(FractionField(QQ['q','t']))
sage: P = Sym.macdonald().P(); P
Symmetric Functions over Fraction Field of Multivariate Polynomial Ring in q, t over Rational Field in the Macdonald P basis
sage: P[2]
McdP[2]

Python

>>> from sage.all import *
>>> Sym = SymmetricFunctions(FractionField(QQ['q','t']))
>>> P = Sym.macdonald().P(); P
Symmetric Functions over Fraction Field of Multivariate Polynomial Ring in q, t over Rational Field in the Macdonald P basis
>>> P[Integer(2)]
McdP[2]

The \(P\) Macdonald basis is upper triangularly related to the monomial symmetric functions and are orthogonal with respect to the \(qt\)-Hall scalar product:

Sage

sage: Sym = SymmetricFunctions(FractionField(QQ['q','t']))
sage: P = Sym.macdonald().P(); P
Symmetric Functions over Fraction Field of Multivariate Polynomial Ring in q, t over Rational Field in the Macdonald P basis
sage: m = Sym.monomial()
sage: P.transition_matrix(m,2)
[                          1 (q*t - q + t - 1)/(q*t - 1)]
[                          0                           1]
sage: P([1,1]).scalar_qt(P([2]))
0
sage: P([2]).scalar_qt(P([2]))
(-q^3 + q^2 + q - 1)/(-q*t^2 + q*t + t - 1)
sage: P([1,1]).scalar_qt(P([1,1]))
(-q^2*t + q*t + q - 1)/(-t^3 + t^2 + t - 1)

Python

>>> from sage.all import *
>>> Sym = SymmetricFunctions(FractionField(QQ['q','t']))
>>> P = Sym.macdonald().P(); P
Symmetric Functions over Fraction Field of Multivariate Polynomial Ring in q, t over Rational Field in the Macdonald P basis
>>> m = Sym.monomial()
>>> P.transition_matrix(m,Integer(2))
[                          1 (q*t - q + t - 1)/(q*t - 1)]
[                          0                           1]
>>> P([Integer(1),Integer(1)]).scalar_qt(P([Integer(2)]))
0
>>> P([Integer(2)]).scalar_qt(P([Integer(2)]))
(-q^3 + q^2 + q - 1)/(-q*t^2 + q*t + t - 1)
>>> P([Integer(1),Integer(1)]).scalar_qt(P([Integer(1),Integer(1)]))
(-q^2*t + q*t + q - 1)/(-t^3 + t^2 + t - 1)

When \(q = 0\), the Macdonald polynomials on the \(P\) basis are the same as the Hall-Littlewood polynomials on the \(P\) basis.

Sage

sage: Sym = SymmetricFunctions(FractionField(QQ['t']))
sage: P = Sym.macdonald(q=0).P(); P
Symmetric Functions over Fraction Field of Univariate Polynomial Ring in t over Rational Field in the Macdonald P with q=0 basis
sage: P([2])^2
(t+1)*McdP[2, 2] + (-t+1)*McdP[3, 1] + McdP[4]
sage: HLP = Sym.hall_littlewood().P()
sage: HLP([2])^2
(t+1)*HLP[2, 2] + (-t+1)*HLP[3, 1] + HLP[4]

Python

>>> from sage.all import *
>>> Sym = SymmetricFunctions(FractionField(QQ['t']))
>>> P = Sym.macdonald(q=Integer(0)).P(); P
Symmetric Functions over Fraction Field of Univariate Polynomial Ring in t over Rational Field in the Macdonald P with q=0 basis
>>> P([Integer(2)])**Integer(2)
(t+1)*McdP[2, 2] + (-t+1)*McdP[3, 1] + McdP[4]
>>> HLP = Sym.hall_littlewood().P()
>>> HLP([Integer(2)])**Integer(2)
(t+1)*HLP[2, 2] + (-t+1)*HLP[3, 1] + HLP[4]

Coercions from the \(Q\) and \(J\) basis (proportional) are implemented:

Sage

sage: Sym = SymmetricFunctions(FractionField(QQ['q','t']))
sage: P = Sym.macdonald().P()
sage: Q = Sym.macdonald().Q()
sage: J = Sym.macdonald().J()
sage: s = Sym.schur()

Python

>>> from sage.all import *
>>> Sym = SymmetricFunctions(FractionField(QQ['q','t']))
>>> P = Sym.macdonald().P()
>>> Q = Sym.macdonald().Q()
>>> J = Sym.macdonald().J()
>>> s = Sym.schur()

Sage

sage: P(Q([2]))
((q*t^2-q*t-t+1)/(q^3-q^2-q+1))*McdP[2]
sage: P(Q([2,1]))
((-q*t^4+2*q*t^3-q*t^2+t^2-2*t+1)/(-q^4*t+2*q^3*t-q^2*t+q^2-2*q+1))*McdP[2, 1]

Python

>>> from sage.all import *
>>> P(Q([Integer(2)]))
((q*t^2-q*t-t+1)/(q^3-q^2-q+1))*McdP[2]
>>> P(Q([Integer(2),Integer(1)]))
((-q*t^4+2*q*t^3-q*t^2+t^2-2*t+1)/(-q^4*t+2*q^3*t-q^2*t+q^2-2*q+1))*McdP[2, 1]

Sage

sage: P(J([2]))
(q*t^2-q*t-t+1)*McdP[2]
sage: P(J([2,1]))
(-q*t^4+2*q*t^3-q*t^2+t^2-2*t+1)*McdP[2, 1]

Python

>>> from sage.all import *
>>> P(J([Integer(2)]))
(q*t^2-q*t-t+1)*McdP[2]
>>> P(J([Integer(2),Integer(1)]))
(-q*t^4+2*q*t^3-q*t^2+t^2-2*t+1)*McdP[2, 1]

By transitivity, one get coercions from the classical bases:

Sage

sage: P(s([2]))
((q-t)/(q*t-1))*McdP[1, 1] + McdP[2]
sage: P(s([2,1]))
((q*t-t^2+q-t)/(q*t^2-1))*McdP[1, 1, 1] + McdP[2, 1]

Python

>>> from sage.all import *
>>> P(s([Integer(2)]))
((q-t)/(q*t-1))*McdP[1, 1] + McdP[2]
>>> P(s([Integer(2),Integer(1)]))
((q*t-t^2+q-t)/(q*t^2-1))*McdP[1, 1, 1] + McdP[2, 1]

Sage

sage: Sym = SymmetricFunctions(QQ['x','y','z'].fraction_field())
sage: (x,y,z) = Sym.base_ring().gens()
sage: Macxy = Sym.macdonald(q=x,t=y)
sage: Macyz = Sym.macdonald(q=y,t=z)
sage: Maczx = Sym.macdonald(q=z,t=x)
sage: P1 = Macxy.P()
sage: P2 = Macyz.P()
sage: P3 = Maczx.P()
sage: m(P1[2,1])
((-2*x*y^2+x*y-y^2+x-y+2)/(-x*y^2+1))*m[1, 1, 1] + m[2, 1]
sage: m(P2[2,1])
((-2*y*z^2+y*z-z^2+y-z+2)/(-y*z^2+1))*m[1, 1, 1] + m[2, 1]
sage: m(P1(P2(P3[2,1])))
((-2*x^2*z-x^2+x*z-x+z+2)/(-x^2*z+1))*m[1, 1, 1] + m[2, 1]
sage: P1(P2[2])
((-x*y^2+2*x*y*z-y^2*z-x+2*y-z)/(x*y^2*z-x*y-y*z+1))*McdP[1, 1] + McdP[2]
sage: m(z*P1[2]+x*P2[2])
((x^2*y^2*z+x*y^2*z^2-x^2*y^2+x^2*y*z-x*y*z^2+y^2*z^2-x^2*y-2*x*y*z-y*z^2+x*y-y*z+x+z)/(x*y^2*z-x*y-y*z+1))*m[1, 1] + (x+z)*m[2]

Python

>>> from sage.all import *
>>> Sym = SymmetricFunctions(QQ['x','y','z'].fraction_field())
>>> (x,y,z) = Sym.base_ring().gens()
>>> Macxy = Sym.macdonald(q=x,t=y)
>>> Macyz = Sym.macdonald(q=y,t=z)
>>> Maczx = Sym.macdonald(q=z,t=x)
>>> P1 = Macxy.P()
>>> P2 = Macyz.P()
>>> P3 = Maczx.P()
>>> m(P1[Integer(2),Integer(1)])
((-2*x*y^2+x*y-y^2+x-y+2)/(-x*y^2+1))*m[1, 1, 1] + m[2, 1]
>>> m(P2[Integer(2),Integer(1)])
((-2*y*z^2+y*z-z^2+y-z+2)/(-y*z^2+1))*m[1, 1, 1] + m[2, 1]
>>> m(P1(P2(P3[Integer(2),Integer(1)])))
((-2*x^2*z-x^2+x*z-x+z+2)/(-x^2*z+1))*m[1, 1, 1] + m[2, 1]
>>> P1(P2[Integer(2)])
((-x*y^2+2*x*y*z-y^2*z-x+2*y-z)/(x*y^2*z-x*y-y*z+1))*McdP[1, 1] + McdP[2]
>>> m(z*P1[Integer(2)]+x*P2[Integer(2)])
((x^2*y^2*z+x*y^2*z^2-x^2*y^2+x^2*y*z-x*y*z^2+y^2*z^2-x^2*y-2*x*y*z-y*z^2+x*y-y*z+x+z)/(x*y^2*z-x*y-y*z+1))*m[1, 1] + (x+z)*m[2]

Q()[source]#

Returns the Macdonald polynomials on the \(Q\) basis. These are dual to the Macdonald polynomials on the P basis with respect to the \(qt\)-Hall scalar product. The \(Q\) basis is defined to be a normalized form of the \(J\) basis.

INPUT:

self – a family of Macdonald symmetric function bases

OUTPUT:

returns the \(Q\) Macdonald basis of symmetric functions

EXAMPLES:

Sage

sage: Sym = SymmetricFunctions(FractionField(QQ['q','t']))
sage: Q = Sym.macdonald().Q(); Q
Symmetric Functions over Fraction Field of Multivariate Polynomial Ring in q, t over Rational Field in the Macdonald Q basis
sage: P = Sym.macdonald().P()
sage: Q([2]).scalar_qt(P([2]))
1
sage: Q([2]).scalar_qt(P([1,1]))
0
sage: Q([1,1]).scalar_qt(P([2]))
0
sage: Q([1,1]).scalar_qt(P([1,1]))
1
sage: Q(P([2]))
((q^3-q^2-q+1)/(q*t^2-q*t-t+1))*McdQ[2]
sage: Q(P([1,1]))
((q^2*t-q*t-q+1)/(t^3-t^2-t+1))*McdQ[1, 1]

Python

>>> from sage.all import *
>>> Sym = SymmetricFunctions(FractionField(QQ['q','t']))
>>> Q = Sym.macdonald().Q(); Q
Symmetric Functions over Fraction Field of Multivariate Polynomial Ring in q, t over Rational Field in the Macdonald Q basis
>>> P = Sym.macdonald().P()
>>> Q([Integer(2)]).scalar_qt(P([Integer(2)]))
1
>>> Q([Integer(2)]).scalar_qt(P([Integer(1),Integer(1)]))
0
>>> Q([Integer(1),Integer(1)]).scalar_qt(P([Integer(2)]))
0
>>> Q([Integer(1),Integer(1)]).scalar_qt(P([Integer(1),Integer(1)]))
1
>>> Q(P([Integer(2)]))
((q^3-q^2-q+1)/(q*t^2-q*t-t+1))*McdQ[2]
>>> Q(P([Integer(1),Integer(1)]))
((q^2*t-q*t-q+1)/(t^3-t^2-t+1))*McdQ[1, 1]

Coercions from the \(P\) and \(J\) basis (proportional) are implemented:

Sage

sage: P = Sym.macdonald().P()
sage: Q = Sym.macdonald().Q()
sage: J = Sym.macdonald().J()
sage: s = Sym.schur()

Python

>>> from sage.all import *
>>> P = Sym.macdonald().P()
>>> Q = Sym.macdonald().Q()
>>> J = Sym.macdonald().J()
>>> s = Sym.schur()

Sage

sage: Q(J([2]))
(q^3-q^2-q+1)*McdQ[2]

Python

>>> from sage.all import *
>>> Q(J([Integer(2)]))
(q^3-q^2-q+1)*McdQ[2]

Sage

sage: Q(P([2]))
((q^3-q^2-q+1)/(q*t^2-q*t-t+1))*McdQ[2]
sage: P(Q(P([2])))
McdP[2]
sage: Q(P(Q([2])))
McdQ[2]

Python

>>> from sage.all import *
>>> Q(P([Integer(2)]))
((q^3-q^2-q+1)/(q*t^2-q*t-t+1))*McdQ[2]
>>> P(Q(P([Integer(2)])))
McdP[2]
>>> Q(P(Q([Integer(2)])))
McdQ[2]

By transitivity, one gets coercions from the classical bases:

Sage

sage: Q(s([2]))
((q^2-q*t-q+t)/(t^3-t^2-t+1))*McdQ[1, 1] + ((q^3-q^2-q+1)/(q*t^2-q*t-t+1))*McdQ[2]

Python

>>> from sage.all import *
>>> Q(s([Integer(2)]))
((q^2-q*t-q+t)/(t^3-t^2-t+1))*McdQ[1, 1] + ((q^3-q^2-q+1)/(q*t^2-q*t-t+1))*McdQ[2]

S()[source]#

Returns the modified Schur functions defined by the plethystic substitution \(S_{\mu} = s_{\mu}[X(1-t)/(1-q)]\). When the Macdonald polynomials in the J basis are expressed in terms of the modified Schur functions at \(q=0\), the coefficients are \(qt\)-Kostka numbers.

INPUT:

self – a family of Macdonald symmetric function bases

OUTPUT:

returns the \(S\) Macdonald basis of symmetric functions

EXAMPLES:

Sage

sage: Sym = SymmetricFunctions(FractionField(QQ['q','t']))
sage: S = Sym.macdonald().S(); S
Symmetric Functions over Fraction Field of Multivariate Polynomial Ring in q, t over Rational Field in the Macdonald S basis
sage: p = Sym.power()
sage: p(S[2,1])
((1/3*t^3-t^2+t-1/3)/(q^3-3*q^2+3*q-1))*p[1, 1, 1] + ((-1/3*t^3+1/3)/(q^3-1))*p[3]
sage: J = Sym.macdonald().J()
sage: S(J([2]))
(q^3-q^2-q+1)*McdS[2]
sage: S(J([1,1]))
(q^2*t-q*t-q+1)*McdS[1, 1] + (q^2-q*t-q+t)*McdS[2]
sage: S = Sym.macdonald(q=0).S()
sage: S(J[1,1])
McdS[1, 1] + t*McdS[2]
sage: S(J[2])
q*McdS[1, 1] + McdS[2]
sage: p(S[2,1])
(-1/3*t^3+t^2-t+1/3)*p[1, 1, 1] + (1/3*t^3-1/3)*p[3]

sage: from sage.combinat.sf.macdonald import qt_kostka
sage: qt_kostka([2],[1,1])
t
sage: qt_kostka([1,1],[2])
q

Python

>>> from sage.all import *
>>> Sym = SymmetricFunctions(FractionField(QQ['q','t']))
>>> S = Sym.macdonald().S(); S
Symmetric Functions over Fraction Field of Multivariate Polynomial Ring in q, t over Rational Field in the Macdonald S basis
>>> p = Sym.power()
>>> p(S[Integer(2),Integer(1)])
((1/3*t^3-t^2+t-1/3)/(q^3-3*q^2+3*q-1))*p[1, 1, 1] + ((-1/3*t^3+1/3)/(q^3-1))*p[3]
>>> J = Sym.macdonald().J()
>>> S(J([Integer(2)]))
(q^3-q^2-q+1)*McdS[2]
>>> S(J([Integer(1),Integer(1)]))
(q^2*t-q*t-q+1)*McdS[1, 1] + (q^2-q*t-q+t)*McdS[2]
>>> S = Sym.macdonald(q=Integer(0)).S()
>>> S(J[Integer(1),Integer(1)])
McdS[1, 1] + t*McdS[2]
>>> S(J[Integer(2)])
q*McdS[1, 1] + McdS[2]
>>> p(S[Integer(2),Integer(1)])
(-1/3*t^3+t^2-t+1/3)*p[1, 1, 1] + (1/3*t^3-1/3)*p[3]

>>> from sage.combinat.sf.macdonald import qt_kostka
>>> qt_kostka([Integer(2)],[Integer(1),Integer(1)])
t
>>> qt_kostka([Integer(1),Integer(1)],[Integer(2)])
q

Coercions to/from the Schur basis are implemented:

Sage

sage: S = Sym.macdonald().S()
sage: s = Sym.schur()
sage: S(s([2]))
((q^2-q*t-q+t)/(t^3-t^2-t+1))*McdS[1, 1] + ((-q^2*t+q*t+q-1)/(-t^3+t^2+t-1))*McdS[2]
sage: s(S([1,1]))
((-q*t^2+q*t+t-1)/(-q^3+q^2+q-1))*s[1, 1] + ((q*t-t^2-q+t)/(-q^3+q^2+q-1))*s[2]

Python

>>> from sage.all import *
>>> S = Sym.macdonald().S()
>>> s = Sym.schur()
>>> S(s([Integer(2)]))
((q^2-q*t-q+t)/(t^3-t^2-t+1))*McdS[1, 1] + ((-q^2*t+q*t+q-1)/(-t^3+t^2+t-1))*McdS[2]
>>> s(S([Integer(1),Integer(1)]))
((-q*t^2+q*t+t-1)/(-q^3+q^2+q-1))*s[1, 1] + ((q*t-t^2-q+t)/(-q^3+q^2+q-1))*s[2]

base_ring()[source]#

Returns the base ring of the symmetric functions where the Macdonald symmetric functions live

INPUT:

self – a family of Macdonald symmetric function bases

OUTPUT:

the base ring associated to the corresponding symmetric function ring

EXAMPLES:

Sage

sage: Sym = SymmetricFunctions(QQ['q'].fraction_field())
sage: Mac = Sym.macdonald(t=0)
sage: Mac.base_ring()
Fraction Field of Univariate Polynomial Ring in q over Rational Field

Python

>>> from sage.all import *
>>> Sym = SymmetricFunctions(QQ['q'].fraction_field())
>>> Mac = Sym.macdonald(t=Integer(0))
>>> Mac.base_ring()
Fraction Field of Univariate Polynomial Ring in q over Rational Field

symmetric_function_ring()[source]#

Returns the base ring of the symmetric functions where the Macdonald symmetric functions live

INPUT:

self – a family of Macdonald symmetric function bases

OUTPUT:

the symmetric function ring associated to the Macdonald bases

EXAMPLES:

Sage

sage: Mac = SymmetricFunctions(QQ['q'].fraction_field()).macdonald(t=0)
sage: Mac.symmetric_function_ring()
Symmetric Functions over Fraction Field of Univariate Polynomial Ring in q over Rational Field

Python

>>> from sage.all import *
>>> Mac = SymmetricFunctions(QQ['q'].fraction_field()).macdonald(t=Integer(0))
>>> Mac.symmetric_function_ring()
Symmetric Functions over Fraction Field of Univariate Polynomial Ring in q over Rational Field

class sage.combinat.sf.macdonald.MacdonaldPolynomials_generic(macdonald)[source]#

Bases: SymmetricFunctionAlgebra_generic

A class for methods for one of the Macdonald bases of the symmetric functions

INPUT:

self – a Macdonald basis
macdonald – a family of Macdonald symmetric function bases

EXAMPLES:

Sage

sage: Sym = SymmetricFunctions(FractionField(QQ['q,t'])); Sym.rename("Sym"); Sym
Sym
sage: Sym.macdonald().P()
Sym in the Macdonald P basis
sage: Sym.macdonald(t=2).P()
Sym in the Macdonald P with t=2 basis
sage: Sym.rename()

Python

>>> from sage.all import *
>>> Sym = SymmetricFunctions(FractionField(QQ['q,t'])); Sym.rename("Sym"); Sym
Sym
>>> Sym.macdonald().P()
Sym in the Macdonald P basis
>>> Sym.macdonald(t=Integer(2)).P()
Sym in the Macdonald P with t=2 basis
>>> Sym.rename()

class Element[source]#

Bases: SymmetricFunctionAlgebra_generic_Element

nabla(q=None, t=None, power=1)[source]#

Return the value of the nabla operator applied to self.

The eigenvectors of the nabla operator are the Macdonald polynomials in the \(Ht\) basis. For more information see: [BGHT1999].

The operator nabla acts on symmetric functions and has the Macdonald \(Ht\) basis as eigenfunctions and the eigenvalues are \(q^{n(\mu')} t^{n(\mu)}\) where \(n(\mu) = \sum_{i} (i-1) \mu_i\) and \(\mu'\) is the conjugate shape of \(\mu\).

If the parameter power is an integer then it calculates nabla to that integer. The default value of power is 1.

INPUT:

self – an element of a Macdonald basis
q, t – optional parameters to specialize
power – an integer (default: 1)

OUTPUT:

returns the symmetric function of \(\nabla\) acting on self

EXAMPLES:

Sage

sage: Sym = SymmetricFunctions(FractionField(QQ['q','t']))
sage: P = Sym.macdonald().P()
sage: P([1,1]).nabla()
((q^2*t+q*t^2-2*t)/(q*t-1))*McdP[1, 1] + McdP[2]
sage: P([1,1]).nabla(t=1)
((q^2*t+q*t-t-1)/(q*t-1))*McdP[1, 1] + McdP[2]
sage: H = Sym.macdonald().H()
sage: H([1,1]).nabla()
t*McdH[1, 1] + (-t^2+1)*McdH[2]
sage: H([1,1]).nabla(q=1)
((t^2+q-t-1)/(q*t-1))*McdH[1, 1] + ((-t^3+t^2+t-1)/(q*t-1))*McdH[2]
sage: H(0).nabla()
0
sage: H([2,2,1]).nabla(t=1/H.t)
((-q^2)/(-t^4))*McdH[2, 2, 1]
sage: H([2,2,1]).nabla(t=1/H.t,power=-1)
((-t^4)/(-q^2))*McdH[2, 2, 1]

Python

>>> from sage.all import *
>>> Sym = SymmetricFunctions(FractionField(QQ['q','t']))
>>> P = Sym.macdonald().P()
>>> P([Integer(1),Integer(1)]).nabla()
((q^2*t+q*t^2-2*t)/(q*t-1))*McdP[1, 1] + McdP[2]
>>> P([Integer(1),Integer(1)]).nabla(t=Integer(1))
((q^2*t+q*t-t-1)/(q*t-1))*McdP[1, 1] + McdP[2]
>>> H = Sym.macdonald().H()
>>> H([Integer(1),Integer(1)]).nabla()
t*McdH[1, 1] + (-t^2+1)*McdH[2]
>>> H([Integer(1),Integer(1)]).nabla(q=Integer(1))
((t^2+q-t-1)/(q*t-1))*McdH[1, 1] + ((-t^3+t^2+t-1)/(q*t-1))*McdH[2]
>>> H(Integer(0)).nabla()
0
>>> H([Integer(2),Integer(2),Integer(1)]).nabla(t=Integer(1)/H.t)
((-q^2)/(-t^4))*McdH[2, 2, 1]
>>> H([Integer(2),Integer(2),Integer(1)]).nabla(t=Integer(1)/H.t,power=-Integer(1))
((-t^4)/(-q^2))*McdH[2, 2, 1]

c1(part)[source]#

Returns the qt-Hall scalar product between J(part) and P(part).

INPUT:

self – a Macdonald basis
part – a partition

OUTPUT:

returns the \(qt\)-Hall scalar product between J(part) and P(part)

EXAMPLES:

Sage

sage: Sym = SymmetricFunctions(FractionField(QQ['q','t']))
sage: P = Sym.macdonald().P()
sage: P.c1(Partition([2,1]))
-q^4*t + 2*q^3*t - q^2*t + q^2 - 2*q + 1

Python

>>> from sage.all import *
>>> Sym = SymmetricFunctions(FractionField(QQ['q','t']))
>>> P = Sym.macdonald().P()
>>> P.c1(Partition([Integer(2),Integer(1)]))
-q^4*t + 2*q^3*t - q^2*t + q^2 - 2*q + 1

c2(part)[source]#

Returns the \(qt\)-Hall scalar product between J(part) and Q(part).

INPUT:

self – a Macdonald basis
part – a partition

OUTPUT:

returns the \(qt\)-Hall scalar product between J(part) and Q(part)

EXAMPLES:

Sage

sage: Sym = SymmetricFunctions(FractionField(QQ['q','t']))
sage: P = Sym.macdonald().P()
sage: P.c2(Partition([2,1]))
-q*t^4 + 2*q*t^3 - q*t^2 + t^2 - 2*t + 1

Python

>>> from sage.all import *
>>> Sym = SymmetricFunctions(FractionField(QQ['q','t']))
>>> P = Sym.macdonald().P()
>>> P.c2(Partition([Integer(2),Integer(1)]))
-q*t^4 + 2*q*t^3 - q*t^2 + t^2 - 2*t + 1

construction()[source]#

Return a pair (F, R), where F is a SymmetricFunctionsFunctor and \(R\) is a ring, such that F(R) returns self.

EXAMPLES:

Sage

sage: Sym = SymmetricFunctions(FractionField(QQ['q']))
sage: J = Sym.macdonald(t=2).J()
sage: J.construction()
(SymmetricFunctionsFunctor[Macdonald J with t=2],
 Fraction Field of Univariate Polynomial Ring in q over Rational Field)

Python

>>> from sage.all import *
>>> Sym = SymmetricFunctions(FractionField(QQ['q']))
>>> J = Sym.macdonald(t=Integer(2)).J()
>>> J.construction()
(SymmetricFunctionsFunctor[Macdonald J with t=2],
 Fraction Field of Univariate Polynomial Ring in q over Rational Field)

macdonald_family()[source]#

Returns the family of Macdonald bases associated to the basis self

INPUT:

self – a Macdonald basis

OUTPUT:

the family of Macdonald symmetric functions associated to self

EXAMPLES:

Sage

sage: MacP = SymmetricFunctions(QQ['q'].fraction_field()).macdonald(t=0).P()
sage: MacP.macdonald_family()
Macdonald polynomials with t=0 over Fraction Field of Univariate Polynomial Ring in q over Rational Field

Python

>>> from sage.all import *
>>> MacP = SymmetricFunctions(QQ['q'].fraction_field()).macdonald(t=Integer(0)).P()
>>> MacP.macdonald_family()
Macdonald polynomials with t=0 over Fraction Field of Univariate Polynomial Ring in q over Rational Field

product(left, right)[source]#

Multiply an element of the Macdonald symmetric function basis self and another symmetric function

Convert to the Schur basis, do the multiplication there, and convert back to self basis.

INPUT:

self – a Macdonald symmetric function basis
left – an element of the basis self
right – another symmetric function

OUTPUT:

the product of left and right expanded in the basis self

EXAMPLES:

Sage

sage: Mac = SymmetricFunctions(FractionField(QQ['q','t'])).macdonald()
sage: H = Mac.H()
sage: J = Mac.J()
sage: P = Mac.P()
sage: Q = Mac.Q()
sage: Ht = Mac.Ht()
sage: J([1])^2 #indirect doctest
((q-1)/(q*t-1))*McdJ[1, 1] + ((t-1)/(q*t-1))*McdJ[2]
sage: J.product( J[1], J[2] )
((-q^2+1)/(-q^2*t+1))*McdJ[2, 1] + ((-t+1)/(-q^2*t+1))*McdJ[3]
sage: H.product( H[1], H[2] )
((q^2-1)/(q^2*t-1))*McdH[2, 1] + ((-t+1)/(-q^2*t+1))*McdH[3]
sage: P.product( P[1], P[2] )
((-q^3*t^2+q*t^2+q^2-1)/(-q^3*t^2+q^2*t+q*t-1))*McdP[2, 1] + McdP[3]
sage: Q.product(Q[1],Q[2])
McdQ[2, 1] + ((q^2*t-q^2+q*t-q+t-1)/(q^2*t-1))*McdQ[3]
sage: Ht.product(Ht[1],Ht[2])
((q^2-1)/(q^2-t))*McdHt[2, 1] + ((t-1)/(-q^2+t))*McdHt[3]

Python

>>> from sage.all import *
>>> Mac = SymmetricFunctions(FractionField(QQ['q','t'])).macdonald()
>>> H = Mac.H()
>>> J = Mac.J()
>>> P = Mac.P()
>>> Q = Mac.Q()
>>> Ht = Mac.Ht()
>>> J([Integer(1)])**Integer(2) #indirect doctest
((q-1)/(q*t-1))*McdJ[1, 1] + ((t-1)/(q*t-1))*McdJ[2]
>>> J.product( J[Integer(1)], J[Integer(2)] )
((-q^2+1)/(-q^2*t+1))*McdJ[2, 1] + ((-t+1)/(-q^2*t+1))*McdJ[3]
>>> H.product( H[Integer(1)], H[Integer(2)] )
((q^2-1)/(q^2*t-1))*McdH[2, 1] + ((-t+1)/(-q^2*t+1))*McdH[3]
>>> P.product( P[Integer(1)], P[Integer(2)] )
((-q^3*t^2+q*t^2+q^2-1)/(-q^3*t^2+q^2*t+q*t-1))*McdP[2, 1] + McdP[3]
>>> Q.product(Q[Integer(1)],Q[Integer(2)])
McdQ[2, 1] + ((q^2*t-q^2+q*t-q+t-1)/(q^2*t-1))*McdQ[3]
>>> Ht.product(Ht[Integer(1)],Ht[Integer(2)])
((q^2-1)/(q^2-t))*McdHt[2, 1] + ((t-1)/(-q^2+t))*McdHt[3]

class sage.combinat.sf.macdonald.MacdonaldPolynomials_h(macdonald)[source]#

Bases: MacdonaldPolynomials_generic

The \(H\) basis is defined as \(H_\mu = \sum_{\lambda} K_{\lambda\mu}(q,t) s_\lambda\) where \(K_{\lambda\mu}(q,t)\) are the Macdonald Kostka coefficients.

In this implementation, it is calculated by using the Macdonald \(Ht\) basis and substituting \(t \rightarrow 1/t\) and multiplying by \(t^{n(\mu)}\).

INPUT:

self – a Macdonald \(H\) basis
macdonald – a family of Macdonald bases

class Element[source]#: Bases: Element

class sage.combinat.sf.macdonald.MacdonaldPolynomials_ht(macdonald)[source]#

Bases: MacdonaldPolynomials_generic

The \(Ht\) basis is defined as \({\tilde H}_\mu = t^{n(\mu)} \sum_{\lambda} K_{\lambda\mu}(q,t^{-1}) s_\lambda\) where \(K_{\lambda\mu}(q,t)\) are the Macdonald \((q,t)\)-Kostka coefficients and \(n(\mu) = \sum_{i} (i-1) \mu_i\).

It is implemented here by using a Pieri formula due to F. Bergeron and M. Haiman [BH2013].

INPUT:

self – a Macdonald \(Ht\) basis
macdonald – a family of Macdonald bases

class Element[source]#

Bases: Element

nabla(q=None, t=None, power=1)[source]#

Returns the value of the nabla operator applied to self. The eigenvectors of the \(nabla\) operator are the Macdonald polynomials in the \(Ht\) basis. For more information see: [BGHT1999].

The operator \(nabla\) acts on symmetric functions and has the Macdonald \(Ht\) basis as eigenfunctions and the eigenvalues are \(q^{n(\mu')} t^{n(\mu)}\) where \(n(\mu) = \sum_{i} (i-1) \mu_i\).

If the parameter power is an integer then it calculates nabla to that integer. The default value of power is 1.

INPUT:

self – an element of the Macdonald \(Ht\) basis
q, t – optional parameters to specialize
power – an integer (default: 1)

OUTPUT:

returns the symmetric function of \(\nabla\) acting on self

EXAMPLES:

Sage

sage: Sym = SymmetricFunctions(FractionField(QQ['q','t']))
sage: Ht = Sym.macdonald().Ht()
sage: t = Ht.t; q = Ht.q
sage: s = Sym.schur()
sage: a = sum(Ht(p) for p in Partitions(3))
sage: Ht(0).nabla()
0
sage: a.nabla() == t^3*Ht([1,1,1])+q*t*Ht([2,1]) + q^3*Ht([3])
True
sage: a.nabla(t=3) == 27*Ht([1,1,1])+3*q*Ht([2,1]) + q^3*Ht([3])
True
sage: a.nabla(q=3) == t^3*Ht([1,1,1])+3*t*Ht([2,1]) + 27*Ht([3])
True
sage: Ht[2,1].nabla(power=-1)
1/(q*t)*McdHt[2, 1]
sage: Ht[2,1].nabla(power=4)
q^4*t^4*McdHt[2, 1]
sage: s(a.nabla(q=3))
(t^6+27*q^3+3*q*t^2)*s[1, 1, 1] + (t^5+t^4+27*q^2+3*q*t+3*t^2+27*q)*s[2, 1] + (t^3+3*t+27)*s[3]
sage: Ht = Sym.macdonald(q=3).Ht()
sage: a = sum(Ht(p) for p in Partitions(3))
sage: s(a.nabla())
(t^6+9*t^2+729)*s[1, 1, 1] + (t^5+t^4+3*t^2+9*t+324)*s[2, 1] + (t^3+3*t+27)*s[3]

Python

>>> from sage.all import *
>>> Sym = SymmetricFunctions(FractionField(QQ['q','t']))
>>> Ht = Sym.macdonald().Ht()
>>> t = Ht.t; q = Ht.q
>>> s = Sym.schur()
>>> a = sum(Ht(p) for p in Partitions(Integer(3)))
>>> Ht(Integer(0)).nabla()
0
>>> a.nabla() == t**Integer(3)*Ht([Integer(1),Integer(1),Integer(1)])+q*t*Ht([Integer(2),Integer(1)]) + q**Integer(3)*Ht([Integer(3)])
True
>>> a.nabla(t=Integer(3)) == Integer(27)*Ht([Integer(1),Integer(1),Integer(1)])+Integer(3)*q*Ht([Integer(2),Integer(1)]) + q**Integer(3)*Ht([Integer(3)])
True
>>> a.nabla(q=Integer(3)) == t**Integer(3)*Ht([Integer(1),Integer(1),Integer(1)])+Integer(3)*t*Ht([Integer(2),Integer(1)]) + Integer(27)*Ht([Integer(3)])
True
>>> Ht[Integer(2),Integer(1)].nabla(power=-Integer(1))
1/(q*t)*McdHt[2, 1]
>>> Ht[Integer(2),Integer(1)].nabla(power=Integer(4))
q^4*t^4*McdHt[2, 1]
>>> s(a.nabla(q=Integer(3)))
(t^6+27*q^3+3*q*t^2)*s[1, 1, 1] + (t^5+t^4+27*q^2+3*q*t+3*t^2+27*q)*s[2, 1] + (t^3+3*t+27)*s[3]
>>> Ht = Sym.macdonald(q=Integer(3)).Ht()
>>> a = sum(Ht(p) for p in Partitions(Integer(3)))
>>> s(a.nabla())
(t^6+9*t^2+729)*s[1, 1, 1] + (t^5+t^4+3*t^2+9*t+324)*s[2, 1] + (t^3+3*t+27)*s[3]

class sage.combinat.sf.macdonald.MacdonaldPolynomials_j(macdonald)[source]#

Bases: MacdonaldPolynomials_generic

The \(J\) basis is calculated using determinantal formulas of Lapointe-Lascoux-Morse giving the action on the \(S\)-basis.

INPUT:

self – a Macdonald \(J\) basis
macdonald – a family of Macdonald bases

class Element[source]#: Bases: Element

class sage.combinat.sf.macdonald.MacdonaldPolynomials_p(macdonald)[source]#

Bases: MacdonaldPolynomials_generic

The \(P\) basis is defined here as the \(J\) basis times a normalizing coefficient \(c2\).

INPUT:

self – a Macdonald \(P\) basis
macdonald – a family of Macdonald bases

class Element[source]#: Bases: Element

scalar_qt_basis(part1, part2=None)[source]#

Returns the scalar product of \(P(part1)\) and \(P(part2)\) This scalar product formula is given in equation (4.11) p.323 and (6.19) p.339 of Macdonald’s book [Mac1995].

INPUT:

self – a Macdonald \(P\) basis
part1, part2 – partitions

OUTPUT:

returns the scalar product of P(part1) and P(part2)

EXAMPLES:

Sage

sage: Sym = SymmetricFunctions(FractionField(QQ['q','t']))
sage: P = Sym.macdonald().P()
sage: P.scalar_qt_basis(Partition([2,1]), Partition([1,1,1]))
0
sage: f = P.scalar_qt_basis(Partition([3,2,1]), Partition([3,2,1]))
sage: factor(f.numerator())
(q - 1)^3 * (q^2*t - 1)^2 * (q^3*t^2 - 1)
sage: factor(f.denominator())
(t - 1)^3 * (q*t^2 - 1)^2 * (q^2*t^3 - 1)

Python

>>> from sage.all import *
>>> Sym = SymmetricFunctions(FractionField(QQ['q','t']))
>>> P = Sym.macdonald().P()
>>> P.scalar_qt_basis(Partition([Integer(2),Integer(1)]), Partition([Integer(1),Integer(1),Integer(1)]))
0
>>> f = P.scalar_qt_basis(Partition([Integer(3),Integer(2),Integer(1)]), Partition([Integer(3),Integer(2),Integer(1)]))
>>> factor(f.numerator())
(q - 1)^3 * (q^2*t - 1)^2 * (q^3*t^2 - 1)
>>> factor(f.denominator())
(t - 1)^3 * (q*t^2 - 1)^2 * (q^2*t^3 - 1)

With a single argument, takes \(part2 = part1\):

Sage

sage: P.scalar_qt_basis(Partition([2,1]), Partition([2,1]))
(-q^4*t + 2*q^3*t - q^2*t + q^2 - 2*q + 1)/(-q*t^4 + 2*q*t^3 - q*t^2 + t^2 - 2*t + 1)

Python

>>> from sage.all import *
>>> P.scalar_qt_basis(Partition([Integer(2),Integer(1)]), Partition([Integer(2),Integer(1)]))
(-q^4*t + 2*q^3*t - q^2*t + q^2 - 2*q + 1)/(-q*t^4 + 2*q*t^3 - q*t^2 + t^2 - 2*t + 1)

class sage.combinat.sf.macdonald.MacdonaldPolynomials_q(macdonald)[source]#

Bases: MacdonaldPolynomials_generic

The \(Q\) basis is defined here as the \(J\) basis times a normalizing coefficient.

INPUT:

self – a Macdonald \(Q\) basis
macdonald – a family of Macdonald bases

class Element[source]#: Bases: Element

class sage.combinat.sf.macdonald.MacdonaldPolynomials_s(macdonald)[source]#

Bases: MacdonaldPolynomials_generic

An implementation of the basis \(s_\lambda[(1-t)X/(1-q)]\)

This is perhaps misnamed as a ‘Macdonald’ basis for the symmetric functions but is used in the calculation of the Macdonald \(J\) basis (see method ‘creation’ below) but does use both of the two parameters and can be specialized to \(s_\lambda[(1-t)X]\) and \(s_\lambda[X/(1-t)]\).

INPUT:

self – a Macdonald \(S\) basis
macdonald – a family of Macdonald bases

class Element[source]#

Bases: Element

creation(k)[source]#

This function is a creation operator for the J-basis for which the action is known on the ‘Macdonald’ S-basis by formula from [LLM1998].

INPUT:

self – an element of the Macdonald \(S\) basis
k – a positive integer

OUTPUT:

returns the column adding operator on the \(J\) basis on self

EXAMPLES:

Sage

sage: Sym = SymmetricFunctions(FractionField(QQ['q','t']))
sage: S = Sym.macdonald().S()
sage: a = S(1)
sage: a.creation(1)
(-q+1)*McdS[1]
sage: a.creation(2)
(q^2*t-q*t-q+1)*McdS[1, 1] + (q^2-q*t-q+t)*McdS[2]

Python

>>> from sage.all import *
>>> Sym = SymmetricFunctions(FractionField(QQ['q','t']))
>>> S = Sym.macdonald().S()
>>> a = S(Integer(1))
>>> a.creation(Integer(1))
(-q+1)*McdS[1]
>>> a.creation(Integer(2))
(q^2*t-q*t-q+1)*McdS[1, 1] + (q^2-q*t-q+t)*McdS[2]

product(left, right)[source]#

The multiplication of the modified Schur functions behaves the same as the multiplication of the Schur functions.

INPUT:

self – a Macdonald \(S\) basis
left, right – a symmetric functions

OUTPUT:

the product of left and right

EXAMPLES:

Sage

sage: Sym = SymmetricFunctions(FractionField(QQ['q','t']))
sage: S = Sym.macdonald().S()
sage: S([2])^2 #indirect doctest
McdS[2, 2] + McdS[3, 1] + McdS[4]

Python

>>> from sage.all import *
>>> Sym = SymmetricFunctions(FractionField(QQ['q','t']))
>>> S = Sym.macdonald().S()
>>> S([Integer(2)])**Integer(2) #indirect doctest
McdS[2, 2] + McdS[3, 1] + McdS[4]

sage.combinat.sf.macdonald.c1(part, q, t)[source]#

This function returns the qt-Hall scalar product between J(part) and P(part).

This coefficient is \(c_\lambda\) in equation (8.1’) p. 352 of Macdonald’s book [Mac1995].

INPUT:

part – a partition
q, t – parameters

OUTPUT:

returns a polynomial of the scalar product between the \(J\) and \(P\) bases

EXAMPLES:

Sage

sage: from sage.combinat.sf.macdonald import c1
sage: R.<q,t> = QQ[]
sage: c1(Partition([2,1]),q,t)
-q^4*t + 2*q^3*t - q^2*t + q^2 - 2*q + 1
sage: c1(Partition([1,1]),q,t)
q^2*t - q*t - q + 1

Python

>>> from sage.all import *
>>> from sage.combinat.sf.macdonald import c1
>>> R = QQ['q, t']; (q, t,) = R._first_ngens(2)
>>> c1(Partition([Integer(2),Integer(1)]),q,t)
-q^4*t + 2*q^3*t - q^2*t + q^2 - 2*q + 1
>>> c1(Partition([Integer(1),Integer(1)]),q,t)
q^2*t - q*t - q + 1

sage.combinat.sf.macdonald.c2(part, q, t)[source]#

This function returns the qt-Hall scalar product between J(part) and Q(part).

This coefficient is \(c_\lambda\) in equation (8.1) p. 352 of Macdonald’s book [Mac1995].

INPUT:

part – a partition
q, t – parameters

OUTPUT:

returns a polynomial of the scalar product between the \(J\) and \(P\) bases

EXAMPLES:

Sage

sage: from sage.combinat.sf.macdonald import c2
sage: R.<q,t> = QQ[]
sage: c2(Partition([1,1]),q,t)
t^3 - t^2 - t + 1
sage: c2(Partition([2,1]),q,t)
-q*t^4 + 2*q*t^3 - q*t^2 + t^2 - 2*t + 1

Python

>>> from sage.all import *
>>> from sage.combinat.sf.macdonald import c2
>>> R = QQ['q, t']; (q, t,) = R._first_ngens(2)
>>> c2(Partition([Integer(1),Integer(1)]),q,t)
t^3 - t^2 - t + 1
>>> c2(Partition([Integer(2),Integer(1)]),q,t)
-q*t^4 + 2*q*t^3 - q*t^2 + t^2 - 2*t + 1

sage.combinat.sf.macdonald.cmunu(nu)[source]#

Return the coefficient of \({\tilde H}_\nu\) in \(h_r^\perp {\tilde H}_\mu\).

Proposition 5 of F. Bergeron and M. Haiman [BH2013] states

\[c_{\mu\nu} = \sum_{\alpha \leftarrow \nu} c_{\mu\alpha} c_{\alpha\nu} B_{\alpha/\nu}/B_{\mu/\nu}\]

where \(c_{\mu\nu}\) is the coefficient of \({\tilde H}_\nu\) in \(h_r^\perp {\tilde H}_\mu\) and \(B_{\mu/\nu}\) is the bi-exponent generator implemented in the function sage.combinat.sf.macdonald.Bmu().

INPUT:

mu, nu – partitions with nu contained in mu

OUTPUT:

an element of the fraction field of polynomials in \(q\) and \(t\)

EXAMPLES:

Sage

sage: from sage.combinat.sf.macdonald import cmunu
sage: cmunu(Partition([2,1]),Partition([1]))
q + t + 1
sage: cmunu(Partition([2,2]),Partition([1,1]))
(-q^3 - q^2 + q*t + t)/(-q + t)
sage: Sym = SymmetricFunctions(QQ['q','t'].fraction_field())
sage: h = Sym.h()
sage: Ht = Sym.macdonald().Ht()
sage: all(Ht[2,2].skew_by(h[r]).coefficient(nu)
....:     == cmunu(Partition([2,2]),nu)
....:     for r in range(1,5) for nu in Partitions(4-r))
True

Python

>>> from sage.all import *
>>> from sage.combinat.sf.macdonald import cmunu
>>> cmunu(Partition([Integer(2),Integer(1)]),Partition([Integer(1)]))
q + t + 1
>>> cmunu(Partition([Integer(2),Integer(2)]),Partition([Integer(1),Integer(1)]))
(-q^3 - q^2 + q*t + t)/(-q + t)
>>> Sym = SymmetricFunctions(QQ['q','t'].fraction_field())
>>> h = Sym.h()
>>> Ht = Sym.macdonald().Ht()
>>> all(Ht[Integer(2),Integer(2)].skew_by(h[r]).coefficient(nu)
...     == cmunu(Partition([Integer(2),Integer(2)]),nu)
...     for r in range(Integer(1),Integer(5)) for nu in Partitions(Integer(4)-r))
True

sage.combinat.sf.macdonald.cmunu1(nu)[source]#

Return the coefficient of \({\tilde H}_\nu\) in \(h_1^\perp {\tilde H}_\mu\).

INPUT:

mu, nu – partitions with nu precedes mu

OUTPUT:

an element of the fraction field of polynomials in \(q\) and \(t\)

EXAMPLES:

Sage

sage: from sage.combinat.sf.macdonald import cmunu1
sage: cmunu1(Partition([2,1]),Partition([2]))
(-t^2 + q)/(q - t)
sage: cmunu1(Partition([2,1]),Partition([1,1]))
(-q^2 + t)/(-q + t)
sage: Sym = SymmetricFunctions(QQ['q','t'].fraction_field())
sage: h = Sym.h()
sage: Ht = Sym.macdonald().Ht()
sage: all(Ht[3,2,1].skew_by(h[1]).coefficient(nu)
....:     == cmunu1(Partition([3,2,1]),nu)
....:     for nu in Partition([3,2,1]).down_list())
True

Python

>>> from sage.all import *
>>> from sage.combinat.sf.macdonald import cmunu1
>>> cmunu1(Partition([Integer(2),Integer(1)]),Partition([Integer(2)]))
(-t^2 + q)/(q - t)
>>> cmunu1(Partition([Integer(2),Integer(1)]),Partition([Integer(1),Integer(1)]))
(-q^2 + t)/(-q + t)
>>> Sym = SymmetricFunctions(QQ['q','t'].fraction_field())
>>> h = Sym.h()
>>> Ht = Sym.macdonald().Ht()
>>> all(Ht[Integer(3),Integer(2),Integer(1)].skew_by(h[Integer(1)]).coefficient(nu)
...     == cmunu1(Partition([Integer(3),Integer(2),Integer(1)]),nu)
...     for nu in Partition([Integer(3),Integer(2),Integer(1)]).down_list())
True

sage.combinat.sf.macdonald.qt_kostka(lam, mu)[source]#

Returns the \(K_{\lambda\mu}(q,t)\) by computing the change of basis from the Macdonald H basis to the Schurs.

INPUT:

lam, mu – partitions of the same size

OUTPUT:

returns the \(q,t\)-Kostka polynomial indexed by the partitions lam and mu

EXAMPLES:

Sage

sage: from sage.combinat.sf.macdonald import qt_kostka
sage: qt_kostka([2,1,1],[1,1,1,1])
t^3 + t^2 + t
sage: qt_kostka([1,1,1,1],[2,1,1])
q
sage: qt_kostka([1,1,1,1],[3,1])
q^3
sage: qt_kostka([1,1,1,1],[1,1,1,1])
1
sage: qt_kostka([2,1,1],[2,2])
q^2*t + q*t + q
sage: qt_kostka([2,2],[2,2])
q^2*t^2 + 1
sage: qt_kostka([4],[3,1])
t
sage: qt_kostka([2,2],[3,1])
q^2*t + q
sage: qt_kostka([3,1],[2,1,1])
q*t^3 + t^2 + t
sage: qt_kostka([2,1,1],[2,1,1])
q*t^2 + q*t + 1
sage: qt_kostka([2,1],[1,1,1,1])
0

Python

>>> from sage.all import *
>>> from sage.combinat.sf.macdonald import qt_kostka
>>> qt_kostka([Integer(2),Integer(1),Integer(1)],[Integer(1),Integer(1),Integer(1),Integer(1)])
t^3 + t^2 + t
>>> qt_kostka([Integer(1),Integer(1),Integer(1),Integer(1)],[Integer(2),Integer(1),Integer(1)])
q
>>> qt_kostka([Integer(1),Integer(1),Integer(1),Integer(1)],[Integer(3),Integer(1)])
q^3
>>> qt_kostka([Integer(1),Integer(1),Integer(1),Integer(1)],[Integer(1),Integer(1),Integer(1),Integer(1)])
1
>>> qt_kostka([Integer(2),Integer(1),Integer(1)],[Integer(2),Integer(2)])
q^2*t + q*t + q
>>> qt_kostka([Integer(2),Integer(2)],[Integer(2),Integer(2)])
q^2*t^2 + 1
>>> qt_kostka([Integer(4)],[Integer(3),Integer(1)])
t
>>> qt_kostka([Integer(2),Integer(2)],[Integer(3),Integer(1)])
q^2*t + q
>>> qt_kostka([Integer(3),Integer(1)],[Integer(2),Integer(1),Integer(1)])
q*t^3 + t^2 + t
>>> qt_kostka([Integer(2),Integer(1),Integer(1)],[Integer(2),Integer(1),Integer(1)])
q*t^2 + q*t + 1
>>> qt_kostka([Integer(2),Integer(1)],[Integer(1),Integer(1),Integer(1),Integer(1)])
0