Hall-Littlewood Polynomials¶

Notation used in the definitions follows mainly [Mac1995].

class sage.combinat.sf.hall_littlewood.HallLittlewood(Sym, t='t')

The family of Hall-Littlewood symmetric function bases.

The Hall-Littlewood symmetric functions are a family of symmetric functions that depend on a parameter $$t$$.

INPUT:

By default the parameter for these functions is $$t$$, and whatever the parameter is, it must be in the base ring.

EXAMPLES:

sage: SymmetricFunctions(QQ).hall_littlewood(1)
Hall-Littlewood polynomials with t=1 over Rational Field
sage: SymmetricFunctions(QQ['t'].fraction_field()).hall_littlewood()
Hall-Littlewood polynomials over Fraction Field of Univariate Polynomial Ring in t over Rational Field
P()

Return the algebra of symmetric functions in the Hall-Littlewood $$P$$ basis. This is the same as the $$HL$$ basis in John Stembridge’s SF examples file.

INPUT:

• self – a class of Hall-Littlewood symmetric function bases

OUTPUT:

The class of the Hall-Littlewood $$P$$ basis.

EXAMPLES:

sage: Sym = SymmetricFunctions(FractionField(QQ['t']))
sage: HLP = Sym.hall_littlewood().P(); HLP
Symmetric Functions over Fraction Field of Univariate Polynomial Ring in t over Rational Field in the Hall-Littlewood P basis
sage: SP = Sym.hall_littlewood(t=-1).P(); SP
Symmetric Functions over Fraction Field of Univariate Polynomial Ring in t over Rational Field in the Hall-Littlewood P with t=-1 basis
sage: s = Sym.schur()
sage: s(HLP([2,1]))
(-t^2-t)*s[1, 1, 1] + s[2, 1]

The Hall-Littlewood polynomials in the $$P$$ basis at $$t = 0$$ are the Schur functions:

sage: Sym = SymmetricFunctions(QQ)
sage: HLP = Sym.hall_littlewood(t=0).P()
sage: s = Sym.schur()
sage: s(HLP([2,1])) == s([2,1])
True

The Hall-Littlewood polynomials in the $$P$$ basis at $$t = 1$$ are the monomial symmetric functions:

sage: Sym = SymmetricFunctions(QQ)
sage: HLP = Sym.hall_littlewood(t=1).P()
sage: m = Sym.monomial()
sage: m(HLP([2,2,1])) == m([2,2,1])
True

We end with some examples of coercions between:

1. Hall-Littlewood $$P$$ basis.
2. Hall-Littlewood polynomials in the $$Q$$ basis
3. Hall-Littlewood polynomials in the $$Q^\prime$$ basis (via the Schurs)
4. Classical symmetric functions
sage: Sym = SymmetricFunctions(FractionField(QQ['t']))
sage: HLP  = Sym.hall_littlewood().P()
sage: HLQ  = Sym.hall_littlewood().Q()
sage: HLQp = Sym.hall_littlewood().Qp()
sage: s = Sym.schur()
sage: p = Sym.power()
sage: HLP(HLQ()) # indirect doctest
(-t+1)*HLP
sage: HLP(HLQp())
t*HLP[1, 1] + HLP
sage: HLP(s())
t*HLP[1, 1] + HLP
sage: HLP(p())
(t-1)*HLP[1, 1] + HLP
sage: s = HLQp.symmetric_function_ring().s()
sage: HLQp.transition_matrix(s,3)
[      1       0       0]
[      t       1       0]
[    t^3 t^2 + t       1]
sage: s.transition_matrix(HLP,3)
[      1       t     t^3]
[      0       1 t^2 + t]
[      0       0       1]

The method sage.combinat.sf.sfa.SymmetricFunctionAlgebra_generic_Element.hl_creation_operator() is a creation operator for the $$Q$$ basis:

sage: HLQp.hl_creation_operator().hl_creation_operator()
HLQp[3, 3, 1]

Transitions between bases with the parameter $$t$$ specialized:

sage: Sym = SymmetricFunctions(FractionField(QQ['y','z']))
sage: (y,z) = Sym.base_ring().gens()
sage: HLy = Sym.hall_littlewood(t=y)
sage: HLz = Sym.hall_littlewood(t=z)
sage: Qpy = HLy.Qp()
sage: Qpz = HLz.Qp()
sage: s = Sym.schur()
sage: s( Qpy[3,1] + z*Qpy[2,2] )
z*s[2, 2] + (y*z+1)*s[3, 1] + (y^2*z+y)*s
sage: s( Qpy[3,1] + y*Qpz[2,2] )
y*s[2, 2] + (y*z+1)*s[3, 1] + (y*z^2+y)*s
sage: s( Qpy[3,1] + y*Qpy[2,2] )
y*s[2, 2] + (y^2+1)*s[3, 1] + (y^3+y)*s

sage: Qy = HLy.Q()
sage: Qz = HLz.Q()
sage: Py = HLy.P()
sage: Pz = HLz.P()
sage: Pz(Qpy[2,1])
(y*z^3+z^2+z)*HLP[1, 1, 1] + (y*z+1)*HLP[2, 1] + y*HLP
sage: Pz(Qz[2,1])
(z^2-2*z+1)*HLP[2, 1]
sage: Qz(Py)
((-y+z)/(z^3-z^2-z+1))*HLQ[1, 1] + (1/(-z+1))*HLQ
sage: Qy(Pz)
((y-z)/(y^3-y^2-y+1))*HLQ[1, 1] + (1/(-y+1))*HLQ
sage: Qy.hall_littlewood_family() == HLy
True
sage: Qy.hall_littlewood_family() == HLz
False
sage: Qz.symmetric_function_ring() == Qy.symmetric_function_ring()
True

sage: Sym = SymmetricFunctions(FractionField(QQ['q']))
sage: q = Sym.base_ring().gen()
sage: HL = Sym.hall_littlewood(t=q)
sage: HLQp = HL.Qp()
sage: HLQ = HL.Q()
sage: HLP = HL.P()
sage: s = Sym.schur()
sage: s(HLQp[3,2].plethysm((1-q)*s))/(1-q)^2
(-q^5-q^4)*s[1, 1, 1, 1, 1] + (q^3+q^2)*s[2, 1, 1, 1] - q*s[2, 2, 1] - q*s[3, 1, 1] + s[3, 2]
sage: s(HLP[3,2])
(-q^5-q^4)*s[1, 1, 1, 1, 1] + (q^3+q^2)*s[2, 1, 1, 1] - q*s[2, 2, 1] - q*s[3, 1, 1] + s[3, 2]

The $$P$$ and $$Q$$-Schur at $$t=-1$$ indexed by strict partitions are a basis for the space algebraically generated by the odd power sum symmetric functions:

sage: Sym = SymmetricFunctions(FractionField(QQ['q']))
sage: SP = Sym.hall_littlewood(t=-1).P()
sage: SQ = Sym.hall_littlewood(t=-1).Q()
sage: p = Sym.power()
sage: SP(SQ[3,2,1])
8*HLP[3, 2, 1]
sage: SP(SQ[2,2,1])
0
sage: p(SP[3,2,1])
1/45*p[1, 1, 1, 1, 1, 1] - 1/9*p[3, 1, 1, 1] - 1/9*p[3, 3] + 1/5*p[5, 1]
sage: SP(p[3,3])
-4*HLP[3, 2, 1] + 2*HLP[4, 2] - 2*HLP[5, 1] + HLP
sage: SQ( SQ*SQ -2*(1-q)*SQ )
HLQ[3, 1] + 2*q*HLQ
Q()

Returns the algebra of symmetric functions in Hall-Littlewood $$Q$$ basis. This is the same as the $$Q$$ basis in John Stembridge’s SF examples file.

More extensive examples can be found in the documentation for the Hall-Littlewood $$P$$ basis.

INPUT:

• self – a class of Hall-Littlewood symmetric function bases

OUTPUT:

• returns the class of the Hall-Littlewood $$Q$$ basis

EXAMPLES:

sage: Sym = SymmetricFunctions(FractionField(QQ['t']))
sage: HLQ = Sym.hall_littlewood().Q(); HLQ
Symmetric Functions over Fraction Field of Univariate Polynomial Ring in t over Rational Field in the Hall-Littlewood Q basis
sage: SQ = SymmetricFunctions(QQ).hall_littlewood(t=-1).Q(); SQ
Symmetric Functions over Rational Field in the Hall-Littlewood Q with t=-1 basis
Qp()

Returns the algebra of symmetric functions in Hall-Littlewood $$Q^\prime$$ (Qp) basis. This is dual to the Hall-Littlewood $$P$$ basis with respect to the standard scalar product.

More extensive examples can be found in the documentation for the Hall-Littlewood P basis.

INPUT:

• self – a class of Hall-Littlewood symmetric function bases

OUTPUT:

• returns the class of the Hall-Littlewood $$Qp$$-basis

EXAMPLES:

sage: Sym = SymmetricFunctions(FractionField(QQ['t']))
sage: HLQp = Sym.hall_littlewood().Qp(); HLQp
Symmetric Functions over Fraction Field of Univariate Polynomial Ring in t over Rational Field in the Hall-Littlewood Qp basis
base_ring()

Returns the base ring of the symmetric functions where the Hall-Littlewood symmetric functions live

INPUT:

• self – a class of Hall-Littlewood symmetric function bases

OUTPUT:

The base ring of the symmetric functions.

EXAMPLES

sage: HL = SymmetricFunctions(QQ['t'].fraction_field()).hall_littlewood(t=1)
sage: HL.base_ring()
Fraction Field of Univariate Polynomial Ring in t over Rational Field
symmetric_function_ring()

The ring of symmetric functions associated to the class of Hall-Littlewood symmetric functions

INPUT:

• self – a class of Hall-Littlewood symmetric function bases

OUTPUT:

• returns the ring of symmetric functions

EXAMPLES

sage: HL = SymmetricFunctions(FractionField(QQ['t'])).hall_littlewood()
sage: HL.symmetric_function_ring()
Symmetric Functions over Fraction Field of Univariate Polynomial Ring in t over Rational Field
class sage.combinat.sf.hall_littlewood.HallLittlewood_generic(hall_littlewood)

A class with methods for working with Hall-Littlewood symmetric functions which are common to all bases.

INPUT:

• self – a Hall-Littlewood symmetric function basis
• hall_littlewood – a class of Hall-Littlewood bases
class Element

Methods for elements of a Hall-Littlewood basis that are common to all bases.

expand(n, alphabet='x')

Expands the symmetric function as a symmetric polynomial in n variables.

INPUT:

• self – an element of a Hall-Littlewood basis
• n – a positive integer
• alphabet – a string representing a variable name (default: ‘x’)

OUTPUT:

• returns a symmetric polynomial of self in n variables

EXAMPLES:

sage: Sym = SymmetricFunctions(FractionField(QQ['t']))
sage: HLP = Sym.hall_littlewood().P()
sage: HLQ = Sym.hall_littlewood().Q()
sage: HLQp = Sym.hall_littlewood().Qp()
sage: HLP().expand(2)
x0^2 + (-t + 1)*x0*x1 + x1^2
sage: HLQ().expand(2)
(-t + 1)*x0^2 + (t^2 - 2*t + 1)*x0*x1 + (-t + 1)*x1^2
sage: HLQp().expand(2)
x0^2 + x0*x1 + x1^2
sage: HLQp().expand(2, 'y')
y0^2 + y0*y1 + y1^2
sage: HLQp().expand(1)
x^2
scalar(x, zee=None)

Returns standard scalar product between self and x.

This is the default implementation that converts both self and x into Schur functions and performs the scalar product that basis.

The Hall-Littlewood $$P$$ basis is dual to the $$Qp$$ basis with respect to this scalar product.

INPUT:

• self – an element of a Hall-Littlewood basis
• x – another symmetric element of the symmetric functions

OUTPUT:

• returns the scalar product between self and x

EXAMPLES:

sage: Sym = SymmetricFunctions(FractionField(QQ['t']))
sage: HLP = Sym.hall_littlewood().P()
sage: HLQ = Sym.hall_littlewood().Q()
sage: HLQp = Sym.hall_littlewood().Qp()
sage: HLP().scalar(HLQp())
1
sage: HLP().scalar(HLQp([1,1]))
0
sage: HLP().scalar(HLQ(), lambda mu: mu.centralizer_size(t = HLP.t))
1
sage: HLP().scalar(HLQ([1,1]), lambda mu: mu.centralizer_size(t = HLP.t))
0
scalar_hl(x, t=None)

Returns the Hall-Littlewood (with parameter t) scalar product of self and x.

The Hall-Littlewood scalar product is defined in Macdonald’s book [Mac1995]. The power sum basis is orthogonal and $$\langle p_\mu, p_\mu \rangle = z_\mu \prod_{i} 1/(1-t^{\mu_i})$$

The Hall-Littlewood $$P$$ basis is dual to the $$Q$$ basis with respect to this scalar product.

INPUT:

• self – an element of a Hall-Littlewood basis
• x – another symmetric element of the symmetric functions
• t – an optional parameter, if this parameter is not specified then the value of the t from the basis is used in the calculation

OUTPUT:

• returns the Hall-Littlewood scalar product between self and x

EXAMPLES:

sage: Sym = SymmetricFunctions(FractionField(QQ['t']))
sage: HLP = Sym.hall_littlewood().P()
sage: HLQ = Sym.hall_littlewood().Q()
sage: HLP().scalar_hl(HLQ())
1
sage: HLP().scalar_hl(HLQ([1,1]))
0
sage: HLQ().scalar_hl(HLQ())
-t + 1
sage: HLQ().scalar_hl(HLQ([1,1]))
0
sage: HLP().scalar_hl(HLP())
-1/(t - 1)
hall_littlewood_family()

The family of Hall-Littlewood bases associated to self

INPUT:

• self – a Hall-Littlewood symmetric function basis

OUTPUT:

• returns the class of Hall-Littlewood bases

EXAMPLES

sage: HLP = SymmetricFunctions(FractionField(QQ['t'])).hall_littlewood(1).P()
sage: HLP.hall_littlewood_family()
Hall-Littlewood polynomials with t=1 over Fraction Field of Univariate Polynomial Ring in t over Rational Field
transition_matrix(basis, n)

Returns the transitions matrix between self and basis for the homogeneous component of degree n.

INPUT:

• self – a Hall-Littlewood symmetric function basis
• basis – another symmetric function basis
• n – a non-negative integer representing the degree

OUTPUT:

• Returns a $$r \times r$$ matrix of elements of the base ring of self where $$r$$ is the number of partitions of n. The entry corresponding to row $$\mu$$, column $$\nu$$ is the coefficient of basis $$(\nu)$$ in self $$(\mu)$$

EXAMPLES:

sage: Sym = SymmetricFunctions(FractionField(QQ['t']))
sage: HLP = Sym.hall_littlewood().P()
sage: s   = Sym.schur()
sage: HLP.transition_matrix(s, 4)
[             1             -t              0            t^2           -t^3]
[             0              1             -t             -t      t^3 + t^2]
[             0              0              1             -t            t^3]
[             0              0              0              1 -t^3 - t^2 - t]
[             0              0              0              0              1]
sage: HLQ = Sym.hall_littlewood().Q()
sage: HLQ.transition_matrix(s,3)
[                        -t + 1                        t^2 - t                     -t^3 + t^2]
[                             0                  t^2 - 2*t + 1           -t^4 + t^3 + t^2 - t]
[                             0                              0 -t^6 + t^5 + t^4 - t^2 - t + 1]
sage: HLQp = Sym.hall_littlewood().Qp()
sage: HLQp.transition_matrix(s,3)
[      1       0       0]
[      t       1       0]
[    t^3 t^2 + t       1]
class sage.combinat.sf.hall_littlewood.HallLittlewood_p(hall_littlewood)

A class representing the Hall-Littlewood $$P$$ basis of symmetric functions

class Element
class sage.combinat.sf.hall_littlewood.HallLittlewood_q(hall_littlewood)

The $$Q$$ basis is defined as a normalization of the $$P$$ basis.

INPUT:

• self – an instance of the Hall-Littlewood $$P$$ basis
• hall_littlewood – a class for the family of Hall-Littlewood bases

EXAMPLES:

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

sage: Sym = SymmetricFunctions(FractionField(QQ['t']))
sage: HLP = Sym.hall_littlewood().P()
sage: HLQ = Sym.hall_littlewood().Q()
sage: HLQp = Sym.hall_littlewood().Qp()
sage: s = Sym.schur(); p = Sym.power()
sage: HLQ( HLP([2,1]) + HLP() )
(1/(t^2-2*t+1))*HLQ[2, 1] - (1/(t-1))*HLQ
sage: HLQ(HLQp()) # indirect doctest
(t/(t^3-t^2-t+1))*HLQ[1, 1] - (1/(t-1))*HLQ
sage: HLQ(s())
(t/(t^3-t^2-t+1))*HLQ[1, 1] - (1/(t-1))*HLQ
sage: HLQ(p())
(1/(t^2-1))*HLQ[1, 1] - (1/(t-1))*HLQ
class Element
class sage.combinat.sf.hall_littlewood.HallLittlewood_qp(hall_littlewood)

The Hall-Littlewood $$Qp$$ basis is calculated through the symmetrica library (see the function HallLittlewood_qp._to_s()).

INPUT:

• self – an instance of the Hall-Littlewood $$P$$ basis
• hall_littlewood – a class for the family of Hall-Littlewood bases

EXAMPLES:

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

sage: Sym = SymmetricFunctions(FractionField(QQ['t']))
sage: HLP = Sym.hall_littlewood().P()
sage: HLQ = Sym.hall_littlewood().Q()
sage: HLQp = Sym.hall_littlewood().Qp()
sage: s = Sym.schur(); p = Sym.power()
sage: HLQp(HLP()) # indirect doctest
-t*HLQp[1, 1] + (t^2+1)*HLQp
sage: HLQp(s(HLQ())) # work around bug reported in ticket #12969
(t^2-t)*HLQp[1, 1] + (-t^3+t^2-t+1)*HLQp
sage: HLQp(s())
HLQp
sage: HLQp(p())
-HLQp[1, 1] + (t+1)*HLQp
sage: s = HLQp.symmetric_function_ring().s()
sage: HLQp.transition_matrix(s,3)
[      1       0       0]
[      t       1       0]
[    t^3 t^2 + t       1]
sage: s.transition_matrix(HLP,3)
[      1       t     t^3]
[      0       1 t^2 + t]
[      0       0       1]
class Element