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r""" Quotient of symmetric function space by ideal generated by Hall-Littlewood symmetric functions
The quotient of symmetric functions by the ideal generated by the Hall-Littlewood P symmetric functions indexed by partitions with first part greater than `k`. When `t=1` this space is the quotient of the symmetric functions by the ideal generated by the monomial symmetric functions indexed by partitions with first part greater than `k`.
AUTHORS:
- Chris Berg (2012-12-01)
- Mike Zabrocki - `k`-bounded Hall Littlewood P and dual `k`-Schur functions (2012-12-02) """ #***************************************************************************** # Copyright (C) 2012 Chris Berg <chrisjamesberg@gmail.com> # Based off of similar code of Jason Bandlow, Anne Schilling # and Mike Zabrocki # # Distributed under the terms of the GNU General Public License (GPL) # # This code is distributed in the hope that it will be useful, # but WITHOUT ANY WARRANTY; without even the implied warranty of # MERCHANTABILITY or FITNESS FOR A PARTICULAR PURPOSE. See the GNU # General Public License for more details. # # The full text of the GPL is available at: # # http://www.gnu.org/licenses/ #*****************************************************************************
r""" Initialization of the ring of Symmetric functions modulo the ideal of monomial symmetric functions which are indexed by partitions whose first part is greater than `k`.
INPUT:
- ``Sym`` -- an element of class :class:`sage.combinat.sf.sf.SymmetricFunctions`
- ``k`` -- a positive integer
- ``R`` -- a ring
EXAMPLES::
sage: Sym = SymmetricFunctions(QQ) sage: Q = Sym.kBoundedQuotient(3,t=1) sage: Q 3-Bounded Quotient of Symmetric Functions over Rational Field with t=1 sage: km = Q.km() sage: km 3-Bounded Quotient of Symmetric Functions over Rational Field with t=1 in the 3-bounded monomial basis sage: F = Q.affineSchur() sage: F(km(F[3,1,1])) == F[3,1,1] True sage: km(F(km([3,2]))) == km[3,2] True sage: F[3,2].lift() m[1, 1, 1, 1, 1] + m[2, 1, 1, 1] + m[2, 2, 1] + m[3, 1, 1] + m[3, 2] sage: F[2,1]*F[2,1] 2*F3[1, 1, 1, 1, 1, 1] + 4*F3[2, 1, 1, 1, 1] + 4*F3[2, 2, 1, 1] + 4*F3[2, 2, 2] + 2*F3[3, 1, 1, 1] + 4*F3[3, 2, 1] + 2*F3[3, 3] sage: F[1,2] Traceback (most recent call last): ... ValueError: [1, 2] is not a valid partition sage: F[4,2] Traceback (most recent call last): ... ValueError: Partition is not 3-bounded sage: km[2,1]*km[2,1] 4*m3[2, 2, 1, 1] + 6*m3[2, 2, 2] + 2*m3[3, 2, 1] + 2*m3[3, 3] sage: HLPk = Q.kHallLittlewoodP() sage: HLPk[2,1]*HLPk[2,1] 4*HLP3[2, 2, 1, 1] + 6*HLP3[2, 2, 2] + 2*HLP3[3, 2, 1] + 2*HLP3[3, 3] sage: dks = Q.dual_k_Schur() sage: dks[2,1]*dks[2,1] 2*dks3[1, 1, 1, 1, 1, 1] + 4*dks3[2, 1, 1, 1, 1] + 4*dks3[2, 2, 1, 1] + 4*dks3[2, 2, 2] + 2*dks3[3, 1, 1, 1] + 4*dks3[3, 2, 1] + 2*dks3[3, 3]
::
sage: Q = Sym.kBoundedQuotient(3) Traceback (most recent call last): ... TypeError: unable to convert 't' to a rational sage: Sym = SymmetricFunctions(QQ['t'].fraction_field()) sage: Q = Sym.kBoundedQuotient(3) sage: km = Q.km() sage: F = Q.affineSchur() sage: F(km(F[3,1,1])) == F[3,1,1] True sage: km(F(km([3,2]))) == km[3,2] True sage: dks = Q.dual_k_Schur() sage: HLPk = Q.kHallLittlewoodP() sage: dks(HLPk(dks[3,1,1])) == dks[3,1,1] True sage: km(dks(km([3,2]))) == km[3,2] True sage: dks[2,1]*dks[2,1] (t^3+t^2)*dks3[1, 1, 1, 1, 1, 1] + (2*t^2+2*t)*dks3[2, 1, 1, 1, 1] + (t^2+2*t+1)*dks3[2, 2, 1, 1] + (t^2+2*t+1)*dks3[2, 2, 2] + (t+1)*dks3[3, 1, 1, 1] + (2*t+2)*dks3[3, 2, 1] + (t+1)*dks3[3, 3]
TESTS::
sage: TestSuite(Q).run()
""" else:
r"""
Returns the Symmetric Functions over the same ring as ``self``. This is needed to realize our ring as a quotient.
TESTS::
sage: Sym = SymmetricFunctions(QQ) sage: Q = Sym.kBoundedQuotient(3,t=1) sage: Q.ambient() Symmetric Functions over Rational Field
"""
r""" Returns a particular realization of ``self`` (the basis of `k`-bounded monomials if `t=1` and the basis of `k`-bounded Hall-Littlewood functions otherwise).
EXAMPLES::
sage: Sym = SymmetricFunctions(QQ) sage: Q = Sym.kBoundedQuotient(3,t=1) sage: Q.a_realization() 3-Bounded Quotient of Symmetric Functions over Rational Field with t=1 in the 3-bounded monomial basis sage: Q = Sym.kBoundedQuotient(3,t=2) sage: Q.a_realization() 3-Bounded Quotient of Symmetric Functions over Rational Field with t=2 in the 3-bounded Hall-Littlewood P basis """ else:
r""" Representation of ``self``.
TESTS::
sage: Sym = SymmetricFunctions(RR) # indirect doctest sage: Sym.kBoundedQuotient(4,t=1) 4-Bounded Quotient of Symmetric Functions over Real Field with 53 bits of precision with t=1.00000000000000 """
r""" The monomial basis of the `k`-bounded quotient of symmetric functions, indexed by `k`-bounded partitions.
EXAMPLES::
sage: SymmetricFunctions(QQ).kBoundedQuotient(2,t=1).kmonomial() 2-Bounded Quotient of Symmetric Functions over Rational Field with t=1 in the 2-bounded monomial basis """
r""" The Hall-Littlewood P basis of the `k`-bounded quotient of symmetric functions, indexed by `k`-bounded partitions. At `t=1` this basis is equal to the `k`-bounded monomial basis and calculations will be faster using elements in the `k`-bounded monomial basis (see :meth:`kmonomial`).
EXAMPLES::
sage: SymmetricFunctions(QQ['t'].fraction_field()).kBoundedQuotient(2).kHallLittlewoodP() 2-Bounded Quotient of Symmetric Functions over Fraction Field of Univariate Polynomial Ring in t over Rational Field in the 2-bounded Hall-Littlewood P basis """
r""" The dual `k`-Schur basis of the `k`-bounded quotient of symmetric functions, indexed by `k`-bounded partitions. At `t=1` this is also equal to the affine Schur basis and calculations will be faster using elements in the :meth:`affineSchur` basis.
EXAMPLES::
sage: SymmetricFunctions(QQ['t'].fraction_field()).kBoundedQuotient(2).dual_k_Schur() 2-Bounded Quotient of Symmetric Functions over Fraction Field of Univariate Polynomial Ring in t over Rational Field in the dual 2-Schur basis """
r""" The affine Schur basis of the `k`-bounded quotient of symmetric functions, indexed by `k`-bounded partitions. This is also equal to the affine Stanley symmetric functions (see :meth:`WeylGroups.ElementMethods.stanley_symmetric_function`) indexed by an affine Grassmannian permutation.
EXAMPLES::
sage: SymmetricFunctions(QQ).kBoundedQuotient(2,t=1).affineSchur() 2-Bounded Quotient of Symmetric Functions over Rational Field with t=1 in the 2-bounded affine Schur basis """
def _G_to_km_on_basis_single_level(self, w, m): r""" Returns the `m^{th}` level of the affine Grothendieck polynomial indexed by the affine Permutation ``w``. This code could be significantly sped up if it didn't depend on the Iwahori Hecke algebra code.
INPUT:
- ``w`` -- An affine permutation (an element of the affine type `A` Weyl group).
- ``m`` -- An integer.
OUTPUT:
- An element of the `k`-bounded quotient.
EXAMPLES::
sage: Q = SymmetricFunctions(QQ).kBoundedQuotient(3,t=1) sage: W = WeylGroup(['A',3,1]) sage: Q._G_to_km_on_basis_single_level(W.an_element(), 3) 0 sage: Q._G_to_km_on_basis_single_level(W.an_element(), 4) m3[1, 1, 1, 1] sage: Q._G_to_km_on_basis_single_level(W.an_element(), 5) -4*m3[1, 1, 1, 1, 1]
"""
r""" Returns the affine Grothendieck polynomial indexed by the affine permutation ``w``. Because this belongs to the completion of the algebra, and hence is an infinite sum, it computes only up to those symmetric functions of degree at most ``m``.
INPUT:
- ``w`` -- An affine permutation (an element of the affine type `A` Weyl group).
- ``m`` -- An integer.
OUTPUT:
- An element of the `k`-bounded quotient.
EXAMPLES::
sage: Q = SymmetricFunctions(QQ).kBoundedQuotient(3,t=1) sage: W = WeylGroup(['A',3,1]) sage: Q._AffineGrothendieck(W.an_element(), 5) m3[1, 1, 1, 1] - 4*m3[1, 1, 1, 1, 1] """
def _AffineGrothendieckPolynomial(self, la, m): r""" Returns the affine Grothendieck polynomial indexed by the partition ``la``. Because this belongs to the completion of the algebra, and hence is an infinite sum, it computes only up to those symmetric functions of degree at most ``m``. This method is here to cache the polynomials.
INPUT:
- ``la`` -- A `k`-bounded partition
- ``m`` -- An integer
EXAMPLES::
sage: Q = SymmetricFunctions(QQ).kBoundedQuotient(3,t=1) sage: Q._AffineGrothendieckPolynomial(Partition([2,1]),4) 2*m3[1, 1, 1] - 8*m3[1, 1, 1, 1] + m3[2, 1] - 3*m3[2, 1, 1] - m3[2, 2] """
r""" Returns the affine Grothendieck polynomial indexed by the partition ``la``. Because this belongs to the completion of the algebra, and hence is an infinite sum, it computes only up to those symmetric functions of degree at most ``m``. See :meth:`_AffineGrothendieckPolynomial` for the code.
INPUT:
- ``la`` -- A `k`-bounded partition
- ``m`` -- An integer
EXAMPLES::
sage: Q = SymmetricFunctions(QQ).kBoundedQuotient(3,t=1) sage: Q.AffineGrothendieckPolynomial([2,1],4) 2*m3[1, 1, 1] - 8*m3[1, 1, 1, 1] + m3[2, 1] - 3*m3[2, 1, 1] - m3[2, 2] """
r""" Returns an element of the quotient ring of `k`-bounded symmetric functions. This method is here to make the TestSuite run properly.
EXAMPLES::
sage: Q = SymmetricFunctions(QQ).kBoundedQuotient(3,t=1) sage: Q.an_element() 2*m3[] + 2*m3[1] + 3*m3[2] """
r""" Returns the unit of the quotient ring of `k`-bounded symmetric functions. This method is here to make the TestSuite run properly.
EXAMPLES::
sage: Q = SymmetricFunctions(QQ).kBoundedQuotient(3,t=1) sage: Q.one() m3[] """
r""" Gives the retract map from the symmetric functions to the quotient ring of `k`-bounded symmetric functions. This method is here to make the TestSuite run properly.
INPUT:
- ``la`` -- A partition
OUTPUT:
- The monomial element of the `k`-bounded quotient indexed by ``la``.
EXAMPLES::
sage: Q = SymmetricFunctions(QQ).kBoundedQuotient(3,t=1) sage: Q.retract([2,1]) m3[2, 1] """
r""" Gives the lift map from the quotient ring of `k`-bounded symmetric functions to the symmetric functions. This method is here to make the TestSuite run properly.
INPUT:
- ``la`` -- A `k`-bounded partition
OUTPUT:
- The monomial element or a Hall-Littlewood P element of the symmetric functions indexed by the partition ``la``.
EXAMPLES::
sage: Q = SymmetricFunctions(QQ).kBoundedQuotient(3,t=1) sage: Q.lift([2,1]) m[2, 1] sage: Q = SymmetricFunctions(QQ['t'].fraction_field()).kBoundedQuotient(3) sage: Q.lift([2,1]) HLP[2, 1] """
""" A list of realizations of the `k`-bounded quotient.
EXAMPLES::
sage: kQ = SymmetricFunctions(QQ['t'].fraction_field()).kBoundedQuotient(3) sage: kQ.realizations() [3-Bounded Quotient of Symmetric Functions over Fraction Field of Univariate Polynomial Ring in t over Rational Field in the 3-bounded monomial basis, 3-Bounded Quotient of Symmetric Functions over Fraction Field of Univariate Polynomial Ring in t over Rational Field in the 3-bounded Hall-Littlewood P basis, 3-Bounded Quotient of Symmetric Functions over Fraction Field of Univariate Polynomial Ring in t over Rational Field in the 3-bounded affine Schur basis, 3-Bounded Quotient of Symmetric Functions over Fraction Field of Univariate Polynomial Ring in t over Rational Field in the dual 3-Schur basis] sage: HLP = kQ.ambient().hall_littlewood().P() sage: all( rzn(HLP[3,2,1]).lift() == HLP[3,2,1] for rzn in kQ.realizations()) True sage: kQ = SymmetricFunctions(QQ).kBoundedQuotient(3,1) sage: kQ.realizations() [3-Bounded Quotient of Symmetric Functions over Rational Field with t=1 in the 3-bounded monomial basis, 3-Bounded Quotient of Symmetric Functions over Rational Field with t=1 in the 3-bounded Hall-Littlewood P basis, 3-Bounded Quotient of Symmetric Functions over Rational Field with t=1 in the 3-bounded affine Schur basis, 3-Bounded Quotient of Symmetric Functions over Rational Field with t=1 in the dual 3-Schur basis] sage: m = kQ.ambient().m() sage: all( rzn(m[3,2,1]).lift() == m[3,2,1] for rzn in kQ.realizations()) True """
r""" The category of bases for the `k`-bounded subspace of symmetric functions. """
""" Initialization of the bases of the `k`-bounded subspace.
INPUT:
- ``base`` -- a basis in the `k`-bounded subspace
TESTS::
sage: Sym = SymmetricFunctions(QQ['t']) sage: from sage.combinat.sf.k_dual import KBoundedQuotientBases sage: Q = Sym.kBoundedQuotient(3,t=1) sage: KQB = KBoundedQuotientBases(Q); KQB Category of k bounded quotient bases of 3-Bounded Quotient of Symmetric Functions over Univariate Polynomial Ring in t over Rational Field with t=1 """
r""" The super categories of ``self``.
EXAMPLES::
sage: Sym = SymmetricFunctions(QQ['t']) sage: from sage.combinat.sf.k_dual import KBoundedQuotientBases sage: Q = Sym.kBoundedQuotient(3,t=1) sage: KQB = KBoundedQuotientBases(Q) sage: KQB.super_categories() [Category of realizations of 3-Bounded Quotient of Symmetric Functions over Univariate Polynomial Ring in t over Rational Field with t=1, Join of Category of graded hopf algebras with basis over Univariate Polynomial Ring in t over Rational Field and Category of quotients of algebras over Univariate Polynomial Ring in t over Rational Field] """
r""" Gives the retract map from the symmetric functions to the quotient ring of `k`-bounded symmetric functions. This method is here to make the TestSuite run properly.
INPUT:
- ``la`` -- A partition
OUTPUT:
- The monomial element of the `k`-bounded quotient indexed by ``la``.
EXAMPLES::
sage: Q = SymmetricFunctions(QQ).kBoundedQuotient(3,t=1) sage: Q.retract([2,1]) m3[2, 1] """ kmhlp = self.realization_of().a_realization() return kmhlp.retract(la)
r""" Needed to rewrite the element constructor because of a bug in free_module.py. Ideally :meth:`_element_constructor_` would be inherited from free_module.py, but it allows for bad inputs.
INPUT:
- ``x`` -- a `k`-bounded partition
OUTPUT:
- an element of the `k`-bounded basis
EXAMPLES::
sage: Q = SymmetricFunctions(QQ).kBoundedQuotient(3,t=1) sage: F = Q.affineSchur() sage: F([2,1]) F3[2, 1] sage: F(Partition([4,1])) Traceback (most recent call last): ... TypeError: do not know how to make x (= [4, 1]) an element of self (=3-Bounded Quotient of Symmetric Functions over Rational Field with t=1 in the 3-bounded affine Schur basis) """
#Coerce ints to Integers x = Integer(x) if x == 0: return self.zero() else: raise TypeError("do not know how to make x (= %s) an element of %s"%(x, self)) #x is an element of the basis enumerated set;
r""" Returns the symmetric functions.
EXAMPLES::
sage: km = SymmetricFunctions(QQ).kBoundedQuotient(3,t=1).km() sage: km.ambient() Symmetric Functions over Rational Field """
r""" Implements shorthand for accessing basis elements.
For a basis `X` indexed by partitions, this method allows for `X[[3,2]]` and `X[3,2]` to be equivalent to `X[Partition([3,2])]`.
Due to limitations in Python syntax, one must use `X[[]]` and not `X[]` for the basis element indexed by the empty partition.
EXAMPLES::
sage: F = SymmetricFunctions(QQ).kBoundedQuotient(3,t=1).affineSchur() sage: F[3,2] F3[3, 2] sage: F[[]] F3[] """ assert len(rest) == 0 else: else:
""" Display elements with single brackets.
The default implementation of CombinatorialFreeModule gives double brackets for basis elements indexed by partitions, i.e., `X[[3,2]]`.
EXAMPLES::
sage: F = SymmetricFunctions(QQ).kBoundedQuotient(3,t=1).affineSchur() sage: F[3,2] # indirect doctest F3[3, 2] """
def one_basis(self): r""" Return the basis element indexing ``1``.
EXAMPLES::
sage: F = SymmetricFunctions(QQ).kBoundedQuotient(3,t=1).affineSchur() sage: F.one() # indirect doctest F3[] """
# This is sufficient for degree to work
r""" Return the degree of the basis element indexed by ``b``.
INPUT:
- ``b`` -- a partition
EXAMPLES::
sage: F = SymmetricFunctions(QQ).kBoundedQuotient(3,t=1).affineSchur() sage: F.degree_on_basis(Partition([3,2])) 5 """
r""" The set of `k`-bounded partitions of all non-negative integers.
EXAMPLES::
sage: km = SymmetricFunctions(QQ).kBoundedQuotient(3,t=1).km() sage: km.indices() 3-Bounded Partitions """ return self._kbounded_partitions
r""" Implements the lift map from the basis ``self`` to the monomial basis of symmetric functions.
INPUT:
- ``la`` -- A `k`-bounded partition.
OUTPUT:
- A symmetric function in the monomial basis.
EXAMPLES::
sage: F = SymmetricFunctions(QQ).kBoundedQuotient(3,t=1).affineSchur() sage: F.lift([3,1]) m[1, 1, 1, 1] + m[2, 1, 1] + m[2, 2] + m[3, 1] sage: Sym = SymmetricFunctions(QQ['t'].fraction_field()) sage: dks = Sym.kBoundedQuotient(3).dual_k_Schur() sage: dks.lift([3,1]) t^5*HLP[1, 1, 1, 1] + t^2*HLP[2, 1, 1] + t*HLP[2, 2] + HLP[3, 1] sage: dks = Sym.kBoundedQuotient(3,t=1).dual_k_Schur() sage: dks.lift([3,1]) m[1, 1, 1, 1] + m[2, 1, 1] + m[2, 2] + m[3, 1] """
r""" Returns the product of two elements ``x`` and ``y``.
INPUT:
- ``x``, ``y`` -- Elements of the `k`-bounded quotient of symmetric functions.
OUTPUT:
- A `k`-bounded symmetric function in the dual `k`-Schur function basis
EXAMPLES::
sage: dks3 = SymmetricFunctions(QQ).kBoundedQuotient(3,t=1).dual_k_Schur() sage: dks3.product(dks3[2,1],dks3[1,1]) 2*dks3[1, 1, 1, 1, 1] + 2*dks3[2, 1, 1, 1] + 2*dks3[2, 2, 1] + dks3[3, 1, 1] + dks3[3, 2] sage: dks3.product(dks3[2,1]+dks3[1], dks3[1,1]) dks3[1, 1, 1] + 2*dks3[1, 1, 1, 1, 1] + dks3[2, 1] + 2*dks3[2, 1, 1, 1] + 2*dks3[2, 2, 1] + dks3[3, 1, 1] + dks3[3, 2] sage: dks3.product(dks3[2,1]+dks3[1], dks3([])) dks3[1] + dks3[2, 1] sage: dks3.product(dks3([]), dks3([])) dks3[] sage: dks3.product(dks3([]), dks3([4,1])) Traceback (most recent call last): ... TypeError: do not know how to make x (= [4, 1]) an element of self (=3-Bounded Quotient of Symmetric Functions over Rational Field with t=1 in the dual 3-Schur basis)
::
sage: dks3 = SymmetricFunctions(QQ['t'].fraction_field()).kBoundedQuotient(3).dual_k_Schur() sage: dks3.product(dks3[2,1],dks3[1,1]) (t^2+t)*dks3[1, 1, 1, 1, 1] + (t+1)*dks3[2, 1, 1, 1] + (t+1)*dks3[2, 2, 1] + dks3[3, 1, 1] + dks3[3, 2] sage: dks3.product(dks3[2,1]+dks3[1], dks3[1,1]) dks3[1, 1, 1] + (t^2+t)*dks3[1, 1, 1, 1, 1] + dks3[2, 1] + (t+1)*dks3[2, 1, 1, 1] + (t+1)*dks3[2, 2, 1] + dks3[3, 1, 1] + dks3[3, 2] sage: dks3.product(dks3[2,1]+dks3[1], dks3([])) dks3[1] + dks3[2, 1] sage: dks3.product(dks3([]), dks3([])) dks3[]
::
sage: F = SymmetricFunctions(QQ).kBoundedQuotient(3,t=1).affineSchur() sage: F.product(F[2,1],F[1,1]) 2*F3[1, 1, 1, 1, 1] + 2*F3[2, 1, 1, 1] + 2*F3[2, 2, 1] + F3[3, 1, 1] + F3[3, 2] sage: F.product(F[2,1]+F[1], F[1,1]) F3[1, 1, 1] + 2*F3[1, 1, 1, 1, 1] + F3[2, 1] + 2*F3[2, 1, 1, 1] + 2*F3[2, 2, 1] + F3[3, 1, 1] + F3[3, 2] sage: F.product(F[2,1]+F[1], F([])) F3[1] + F3[2, 1] sage: F.product(F([]), F([])) F3[] sage: F.product(F([]), F([4,1])) Traceback (most recent call last): ... TypeError: do not know how to make x (= [4, 1]) an element of self (=3-Bounded Quotient of Symmetric Functions over Rational Field with t=1 in the 3-bounded affine Schur basis)
::
sage: F = SymmetricFunctions(QQ['t'].fraction_field()).kBoundedQuotient(3).affineSchur() sage: F.product(F[2,1],F[1,1]) 2*F3[1, 1, 1, 1, 1] + 2*F3[2, 1, 1, 1] + 2*F3[2, 2, 1] + F3[3, 1, 1] + F3[3, 2] sage: F.product(F[2,1],F[2]) (t^4+t^3-2*t^2+1)*F3[1, 1, 1, 1, 1] + (-t^2+t+1)*F3[2, 1, 1, 1] + (-t^2+t+2)*F3[2, 2, 1] + (t+1)*F3[3, 1, 1] + (t+1)*F3[3, 2] sage: F.product(F[2,1]+F[1], F[1,1]) F3[1, 1, 1] + 2*F3[1, 1, 1, 1, 1] + F3[2, 1] + 2*F3[2, 1, 1, 1] + 2*F3[2, 2, 1] + F3[3, 1, 1] + F3[3, 2] sage: F.product(F[2,1]+F[1], F([])) F3[1] + F3[2, 1] sage: F.product(F([]), F([])) F3[]
::
sage: km = SymmetricFunctions(QQ).kBoundedQuotient(3,t=1).km() sage: km.product(km[2,1],km[2,1]) 4*m3[2, 2, 1, 1] + 6*m3[2, 2, 2] + 2*m3[3, 2, 1] + 2*m3[3, 3] sage: Q3 = SymmetricFunctions(FractionField(QQ['t'])).kBoundedQuotient(3) sage: km = Q3.km() sage: km.product(km[2,1],km[2,1]) (t^5+7*t^4-8*t^3-28*t^2+47*t-19)*m3[1, 1, 1, 1, 1, 1] + (t^4-3*t^3-9*t^2+23*t-12)*m3[2, 1, 1, 1, 1] + (-t^3-3*t^2+11*t-3)*m3[2, 2, 1, 1] + (-t^2+5*t+2)*m3[2, 2, 2] + (6*t-6)*m3[3, 1, 1, 1] + (3*t-1)*m3[3, 2, 1] + (t+1)*m3[3, 3] sage: dks = Q3.dual_k_Schur() sage: km.product(dks[2,1],dks[1,1]) 20*m3[1, 1, 1, 1, 1] + 9*m3[2, 1, 1, 1] + 4*m3[2, 2, 1] + 2*m3[3, 1, 1] + m3[3, 2] """
r""" Return the antipode of ``element`` via lifting to the symmetric functions and then retracting into the `k`-bounded quotient basis.
INPUT:
- ``element`` -- an element in a basis of the ring of symmetric functions
EXAMPLES::
sage: dks3 = SymmetricFunctions(QQ).kBoundedQuotient(3,t=1).dual_k_Schur() sage: dks3[3,2].antipode() -dks3[1, 1, 1, 1, 1] sage: km = SymmetricFunctions(QQ).kBoundedQuotient(3,t=1).km() sage: km[3,2].antipode() m3[3, 2] sage: km.antipode(km[3,2]) m3[3, 2] sage: m = SymmetricFunctions(QQ).m() sage: m[3,2].antipode() m[3, 2] + 2*m[5]
::
sage: km = SymmetricFunctions(FractionField(QQ['t'])).kBoundedQuotient(3).km() sage: km[1,1,1,1].antipode() (t^3-3*t^2+3*t)*m3[1, 1, 1, 1] + (-t^2+2*t)*m3[2, 1, 1] + t*m3[2, 2] + t*m3[3, 1] sage: kHP = SymmetricFunctions(FractionField(QQ['t'])).kBoundedQuotient(3).kHLP() sage: kHP[2,2].antipode() (t^9-t^6-t^5+t^2)*HLP3[1, 1, 1, 1] + (t^6-t^3-t^2+t)*HLP3[2, 1, 1] + (t^5-t^2+1)*HLP3[2, 2] + (t^4-t)*HLP3[3, 1] sage: dks = SymmetricFunctions(FractionField(QQ['t'])).kBoundedQuotient(3).dks() sage: dks[2,2].antipode() dks3[2, 2] sage: dks[3,2].antipode() -t^2*dks3[1, 1, 1, 1, 1] + (t^2-1)*dks3[2, 2, 1] + (-t^5+t)*dks3[3, 2] """
r""" Return the coproduct of ``element`` via lifting to the symmetric functions and then returning to the `k`-bounded quotient basis. This method is implemented for all `t` but is (weakly) conjectured to not be the correct operation for arbitrary `t` because the coproduct on dual-`k`-Schur functions does not have a positive expansion.
INPUT:
- ``element`` -- an element in a basis of the ring of symmetric functions
EXAMPLES::
sage: Q3 = SymmetricFunctions(QQ).kBoundedQuotient(3,t=1) sage: km = Q3.km() sage: km[3,2].coproduct() m3[] # m3[3, 2] + m3[2] # m3[3] + m3[3] # m3[2] + m3[3, 2] # m3[] sage: dks3 = Q3.dual_k_Schur() sage: dks3[2,2].coproduct() dks3[] # dks3[2, 2] + dks3[1] # dks3[2, 1] + dks3[1, 1] # dks3[1, 1] + dks3[2] # dks3[2] + dks3[2, 1] # dks3[1] + dks3[2, 2] # dks3[]
::
sage: Q3t = SymmetricFunctions(FractionField(QQ['t'])).kBoundedQuotient(3) sage: km = Q3t.km() sage: km[3,2].coproduct() m3[] # m3[3, 2] + m3[2] # m3[3] + m3[3] # m3[2] + m3[3, 2] # m3[] sage: dks = Q3t.dks() sage: dks[2,1,1].coproduct() dks3[] # dks3[2, 1, 1] + (-t+1)*dks3[1] # dks3[1, 1, 1] + dks3[1] # dks3[2, 1] + (-t+1)*dks3[1, 1] # dks3[1, 1] + dks3[1, 1] # dks3[2] + (-t+1)*dks3[1, 1, 1] # dks3[1] + dks3[2] # dks3[1, 1] + dks3[2, 1] # dks3[1] + dks3[2, 1, 1] # dks3[] sage: kHLP = Q3t.kHLP() sage: kHLP[2,1].coproduct() HLP3[] # HLP3[2, 1] + (-t^2+1)*HLP3[1] # HLP3[1, 1] + HLP3[1] # HLP3[2] + (-t^2+1)*HLP3[1, 1] # HLP3[1] + HLP3[2] # HLP3[1] + HLP3[2, 1] # HLP3[] sage: km.coproduct(km[3,2]) m3[] # m3[3, 2] + m3[2] # m3[3] + m3[3] # m3[2] + m3[3, 2] # m3[] """ for ((x,y), coeff) in element.lift().coproduct())
r""" Return the counit of ``element``.
The counit is the constant term of ``element``.
INPUT:
- ``element`` -- an element in a basis
EXAMPLES::
sage: km = SymmetricFunctions(FractionField(QQ['t'])).kBoundedQuotient(3).km() sage: f = 2*km[2,1] - 3*km([]) sage: f.counit() -3 sage: km.counit(f) -3 """
r""" Abstract base class for the bases of the `k`-bounded quotient. """ r""" Initializes ``self``.
INPUT:
- ``kBoundedRing`` -- an element which is of class :class:`KBoundedQuotient` - ``prefix`` -- a string used to distinguish this basis, and used in printing.
EXAMPLES::
sage: from sage.combinat.sf.k_dual import kMonomial sage: km = kMonomial(SymmetricFunctions(QQ).kBoundedQuotient(4,t=1)) sage: km.prefix() # indirect doctest 'm4' sage: isinstance(km, sage.combinat.sf.k_dual.KBoundedQuotientBasis) True
""" kBoundedRing.indices(), category= KBoundedQuotientBases(kBoundedRing), prefix='%s%d'%(prefix, kBoundedRing.k))
# The following are meant to be inherited with the category framework, but # this fails because they are methods of Parent. The trick below overcomes # this problem.
r""" The basis of monomial symmetric functions indexed by partitions with first part less than or equal to `k`. """
r""" Initializes the ring which is the `k`-Bounded monomial quotient basis.
INPUT:
- ``kBoundedRing`` -- an element which is of class :class:`KBoundedQuotient`
EXAMPLES::
sage: from sage.combinat.sf.k_dual import kMonomial sage: km = kMonomial(SymmetricFunctions(QQ).kBoundedQuotient(4,t=1)) sage: km 4-Bounded Quotient of Symmetric Functions over Rational Field with t=1 in the 4-bounded monomial basis sage: TestSuite(km).run() """
""" TESTS::
sage: Sym = SymmetricFunctions(QQ) sage: km = Sym.kBoundedQuotient(3,t=1).km() sage: km._repr_() '3-Bounded Quotient of Symmetric Functions over Rational Field with t=1 in the 3-bounded monomial basis' """
r""" Implements the retract function on the monomial basis. Given a partition ``la``, the retract will return the corresponding `k`-bounded monomial basis element if ``la`` is `k`-bounded; zero otherwise.
INPUT:
- ``la`` -- A partition
OUTPUT:
- A `k`-bounded monomial symmetric function in the `k`-quotient of symmetric functions.
EXAMPLES::
sage: km = SymmetricFunctions(QQ).kBoundedQuotient(3,t=1).km() sage: km.retract(Partition([3,1])) m3[3, 1] sage: km.retract(Partition([4,1])) 0 sage: km.retract([]) m3[] sage: m = SymmetricFunctions(QQ).m() sage: km(m[3, 1]) m3[3, 1] sage: km(m[4, 1]) 0
::
sage: km = SymmetricFunctions(FractionField(QQ['t'])).kBoundedQuotient(3).km() sage: km.retract(Partition([3,1])) m3[3, 1] sage: km.retract(Partition([4,1])) (t^4+t^3-9*t^2+11*t-4)*m3[1, 1, 1, 1, 1] + (-3*t^2+6*t-3)*m3[2, 1, 1, 1] + (-t^2+3*t-2)*m3[2, 2, 1] + (2*t-2)*m3[3, 1, 1] + (t-1)*m3[3, 2] sage: m = SymmetricFunctions(FractionField(QQ['t'])).m() sage: km(m[3, 1]) m3[3, 1] sage: km(m[4, 1]) (t^4+t^3-9*t^2+11*t-4)*m3[1, 1, 1, 1, 1] + (-3*t^2+6*t-3)*m3[2, 1, 1, 1] + (-t^2+3*t-2)*m3[2, 2, 1] + (2*t-2)*m3[3, 1, 1] + (t-1)*m3[3, 2] """ else:
r""" Implements the lift function on the monomial basis. Given a `k`-bounded partition ``la``, the lift will return the corresponding monomial basis element.
INPUT:
- ``la`` -- A `k`-bounded partition
OUTPUT:
- A monomial symmetric function.
EXAMPLES::
sage: km = SymmetricFunctions(QQ).kBoundedQuotient(3,t=1).km() sage: km.lift(Partition([3,1])) m[3, 1] sage: km.lift([]) m[] sage: km.lift(Partition([4,1])) Traceback (most recent call last): ... TypeError: do not know how to make x (= [4, 1]) an element of self (=3-Bounded Quotient of Symmetric Functions over Rational Field with t=1 in the 3-bounded monomial basis) """
r""" The basis of P Hall-Littlewood symmetric functions indexed by partitions with first part less than or equal to `k`. """
r""" Initializes the ring which is the `k`-Bounded Hall-Littlewood P quotient basis.
INPUT:
- ``kBoundedRing`` -- an element which is of class :class:`KBoundedQuotient`
EXAMPLES::
sage: from sage.combinat.sf.k_dual import kbounded_HallLittlewoodP sage: kP = kbounded_HallLittlewoodP(SymmetricFunctions(QQ['t'].fraction_field()).kBoundedQuotient(4)) sage: kP 4-Bounded Quotient of Symmetric Functions over Fraction Field of Univariate Polynomial Ring in t over Rational Field in the 4-bounded Hall-Littlewood P basis sage: TestSuite(kP).run() """
""" TESTS::
sage: Sym = SymmetricFunctions(QQ['t'].fraction_field()) sage: kHLP = Sym.kBoundedQuotient(3).kHallLittlewoodP() sage: kHLP._repr_() '3-Bounded Quotient of Symmetric Functions over Fraction Field of Univariate Polynomial Ring in t over Rational Field in the 3-bounded Hall-Littlewood P basis' """
r""" Converts from the monomial basis to the `k`-bounded Hall-Littlewood P basis. If ``la`` is not `k`-bounded then it returns the projection of the monomial by the ideal generated by the Hall-Littlewood P basis indexed by partitions whose first part is greater than `k`.
INPUT:
- ``la`` - a partition
OUTPUT:
- an element of the `k`-bounded Hall-Littlewood P basis.
EXAMPLES::
sage: Sym = SymmetricFunctions(QQ['t'].fraction_field()) sage: kHLP = Sym.kBoundedQuotient(3).kHallLittlewoodP() sage: kHLP._m_to_kHLP_on_basis([3,1]) (t^5-2*t^2-t+2)*HLP3[1, 1, 1, 1] + (t^2-1)*HLP3[2, 1, 1] + (t-1)*HLP3[2, 2] + HLP3[3, 1] sage: kHLP._m_to_kHLP_on_basis([4]) (t^6-t^5-t^4+t^2+t-1)*HLP3[1, 1, 1, 1] + (t^3-t^2-t+1)*HLP3[2, 1, 1] + (t^2-t)*HLP3[2, 2] + (t-1)*HLP3[3, 1] sage: mk = kHLP.realization_of().km() sage: kHLP(mk([1,1])^2) (t^4+t^3+2*t^2+t+1)*HLP3[1, 1, 1, 1] + (t+1)*HLP3[2, 1, 1] + HLP3[2, 2] sage: kHLP._m_to_kHLP_on_basis([]) HLP3[] sage: kHLP = SymmetricFunctions(QQ).kBoundedQuotient(3,t=1).kHallLittlewoodP() sage: kHLP._m_to_kHLP_on_basis([3,1]) HLP3[3, 1] sage: kHLP._m_to_kHLP_on_basis([4]) 0 sage: mk = kHLP.realization_of().km() sage: kHLP(mk([1,1])^2) 6*HLP3[1, 1, 1, 1] + 2*HLP3[2, 1, 1] + HLP3[2, 2] sage: kHLP(mk([2,1])^2) 4*HLP3[2, 2, 1, 1] + 6*HLP3[2, 2, 2] + 2*HLP3[3, 2, 1] + 2*HLP3[3, 3] """ else: else: if x[0] in self._kbounded_partitions)
r""" Converts from the Hall-Littlewood P basis to the `k`-bounded monomial basis and projects into the `k`-bounded quotient if ``la`` is not a bounded partition.
INPUT:
- ``la`` - a partition
OUTPUT:
- an element of the `k`-bounded monomial basis
EXAMPLES::
sage: Sym = SymmetricFunctions(QQ['t'].fraction_field()) sage: kHLP = Sym.kBoundedQuotient(3).kHallLittlewoodP() sage: kHLP._HLP_to_mk_on_basis([3,1]) (t^3+t^2-5*t+3)*m3[1, 1, 1, 1] + (-2*t+2)*m3[2, 1, 1] + (-t+1)*m3[2, 2] + m3[3, 1] sage: kHLP._HLP_to_mk_on_basis([4,1]) 0 sage: kHLP._HLP_to_mk_on_basis([]) m3[] sage: kHLP = Sym.kBoundedQuotient(3,t=1).kHallLittlewoodP() sage: kHLP._HLP_to_mk_on_basis([3,1]) m3[3, 1] sage: kHLP._HLP_to_mk_on_basis([4,1]) 0 sage: kHLP._HLP_to_mk_on_basis([]) m3[] """ else:
r""" Implements the retract function on the Hall-Littlewood P basis. Given a partition ``la``, the retract will return the corresponding `k`-bounded Hall-Littlewood P basis element if ``la`` is `k`-bounded; zero otherwise.
INPUT:
- ``la`` -- A partition
OUTPUT:
- A `k`-bounded Hall-Littlewood P symmetric function in the `k`-quotient of symmetric functions.
EXAMPLES::
sage: kHLP = SymmetricFunctions(QQ['t'].fraction_field()).kBoundedQuotient(3).kHallLittlewoodP() sage: kHLP.retract(Partition([3,1])) HLP3[3, 1] sage: kHLP.retract(Partition([4,1])) 0 sage: kHLP.retract([]) HLP3[] sage: m = kHLP.realization_of().ambient().m() sage: kHLP(m[2,2]) (t^4-t^3-t+1)*HLP3[1, 1, 1, 1] + (t-1)*HLP3[2, 1, 1] + HLP3[2, 2] """
r""" Implements the lift function on the Hall-Littlewood P basis. Given a `k`-bounded partition ``la``, the lift will return the corresponding Hall-Littlewood P basis element.
INPUT:
- ``la`` -- A `k`-bounded partition
OUTPUT:
- A Hall-Littlewood symmetric function.
EXAMPLES::
sage: kHLP = SymmetricFunctions(QQ['t'].fraction_field()).kBoundedQuotient(3).kHallLittlewoodP() sage: kHLP.lift(Partition([3,1])) HLP[3, 1] sage: kHLP.lift([]) HLP[] sage: kHLP.lift(Partition([4,1])) Traceback (most recent call last): ... TypeError: do not know how to make x (= [4, 1]) an element of self (=3-Bounded Quotient of Symmetric Functions over Fraction Field of Univariate Polynomial Ring in t over Rational Field in the 3-bounded Hall-Littlewood P basis) """
r""" This basis is dual to the `k`-Schur functions. The expansion is given in Section 4.12 of [LLMSSZ]_. When `t=1` this basis is equal to the :class:`AffineSchurFunctions` and that basis is more efficient in this case.
REFERENCES:
.. [LLMSSZ] \T. Lam, L. Lapointe, J. Morse, A. Schilling, M. Shimozono, M. Zabrocki, k-Schur functions and affine Schubert calculus. """
r""" Initializes the ring which is the dual `k`-Schur function basis.
INPUT:
- ``kBoundedRing`` -- an element which is of class :class:`KBoundedQuotient`
EXAMPLES::
sage: from sage.combinat.sf.k_dual import DualkSchurFunctions sage: Sym = SymmetricFunctions(QQ['t'].fraction_field()) sage: dks4 = DualkSchurFunctions(Sym.kBoundedQuotient(4)) sage: dks4 4-Bounded Quotient of Symmetric Functions over Fraction Field of Univariate Polynomial Ring in t over Rational Field in the dual 4-Schur basis sage: TestSuite(dks4).run() # long time (7s on sage.math, 2013) sage: dks4 = DualkSchurFunctions(Sym.kBoundedQuotient(4,t=1)) sage: TestSuite(dks4).run() # long time (7s on sage.math, 2013) """
""" TESTS::
sage: Sym = SymmetricFunctions(QQ['t'].fraction_field()) sage: dks3 = Sym.kBoundedQuotient(3).dual_k_Schur() sage: dks3._repr_() '3-Bounded Quotient of Symmetric Functions over Fraction Field of Univariate Polynomial Ring in t over Rational Field in the dual 3-Schur basis' """
r""" Gives the expansion of the dual `k`-Schur basis element indexed by ``la`` into the Hall-Littlewood P basis.
INPUT:
- ``la`` -- A `k`-bounded partition.
OUTPUT:
- A symmetric function in the Hall-Littlewood P basis
EXAMPLES::
sage: dks3 = SymmetricFunctions(QQ).kBoundedQuotient(3,t=1).dual_k_Schur() sage: dks3._dks_to_khlp_on_basis(Partition([2,1])) 2*HLP3[1, 1, 1] + HLP3[2, 1] sage: dks3._dks_to_khlp_on_basis(Partition([])) HLP3[] sage: dks3._dks_to_khlp_on_basis(Partition([4,1])) Traceback (most recent call last): ... AssertionError: [4, 1] should be an element of 3-Bounded Partitions """
r""" Gives the expansion of the `k`-bounded Hall-Littlewood P basis element indexed by ``la`` into the dual `k`-Schur basis.
INPUT:
- ``la`` -- A `k`-bounded partition.
OUTPUT:
- A `k`-bounded quotient symmetric function in the dual `k`-Schur basis
EXAMPLES::
sage: dks3 = SymmetricFunctions(QQ).kBoundedQuotient(3,t=1).dual_k_Schur() sage: dks3._khlp_to_dks_on_basis(Partition([2,1])) -2*dks3[1, 1, 1] + dks3[2, 1] sage: dks3._khlp_to_dks_on_basis([2,1]) -2*dks3[1, 1, 1] + dks3[2, 1] sage: dks3._khlp_to_dks_on_basis(Partition([])) dks3[] sage: dks3._khlp_to_dks_on_basis(Partition([4,1])) 0
::
sage: dks3 = SymmetricFunctions(QQ['t'].fraction_field()).kBoundedQuotient(3).dual_k_Schur() sage: dks3._khlp_to_dks_on_basis(Partition([2,1])) (-t^2-t)*dks3[1, 1, 1] + dks3[2, 1] sage: dks3._khlp_to_dks_on_basis(Partition([2,2])) (t^3-t^2)*dks3[1, 1, 1, 1] - t*dks3[2, 1, 1] + dks3[2, 2] sage: dks3._khlp_to_dks_on_basis(Partition([])) dks3[] sage: dks3._khlp_to_dks_on_basis(Partition([4,1])) 0 """
r""" This basis is dual to the `k`-Schur functions at `t=1`. This realization follows the monomial expansion given by Lam [Lam2006]_.
REFERENCES:
.. [Lam2006] \T. Lam, Schubert polynomials for the affine Grassmannian, J. Amer. Math. Soc., 21 (2008), 259-281. """
r""" Initializes the ring which is the `k`-Bounded affine Schur quotient basis.
INPUT:
- ``kBoundedRing`` -- an element which is of class :class:`KBoundedQuotient`
EXAMPLES::
sage: from sage.combinat.sf.k_dual import AffineSchurFunctions sage: F = AffineSchurFunctions(SymmetricFunctions(QQ['t']).kBoundedQuotient(4,t=1)) sage: F 4-Bounded Quotient of Symmetric Functions over Univariate Polynomial Ring in t over Rational Field with t=1 in the 4-bounded affine Schur basis sage: TestSuite(F).run() # long time (5s on sage.math, 2013) """
""" TESTS::
sage: Sym = SymmetricFunctions(QQ) sage: F = Sym.kBoundedQuotient(3,t=1).affineSchur() sage: F._repr_() '3-Bounded Quotient of Symmetric Functions over Rational Field with t=1 in the 3-bounded affine Schur basis' """
r""" Gives the expansion of the affine Schur basis element indexed by ``la`` into the monomial basis.
INPUT:
- ``la`` -- A `k`-bounded partition.
OUTPUT:
- A symmetric function in the monomial basis
EXAMPLES::
sage: F = SymmetricFunctions(QQ).kBoundedQuotient(3,t=1).affineSchur() sage: F._F_to_m_on_basis(Partition([2,1])) 2*m[1, 1, 1] + m[2, 1] sage: F._F_to_m_on_basis(Partition([])) m[] sage: km = F.realization_of().km() sage: km( F[2,2] ) 2*m3[1, 1, 1, 1] + m3[2, 1, 1] + m3[2, 2] sage: F._F_to_m_on_basis(Partition([4,1])) Traceback (most recent call last): ... ValueError: the partition must be 3-bounded """
r""" Gives the expansion of the `k`-monomial basis element indexed by ``la`` into the affine Schur basis.
INPUT:
- ``la`` -- A `k`-bounded partition.
OUTPUT:
- A `k`-bounded quotient symmetric function in the affine Schur basis
EXAMPLES::
sage: F = SymmetricFunctions(QQ).kBoundedQuotient(3,t=1).affineSchur() sage: F._m_to_F_on_basis(Partition([2,1])) -2*F3[1, 1, 1] + F3[2, 1] sage: F._m_to_F_on_basis([2,1]) -2*F3[1, 1, 1] + F3[2, 1] sage: F._m_to_F_on_basis(Partition([])) F3[] sage: F._m_to_F_on_basis(Partition([4,1])) Traceback (most recent call last): ... AssertionError: [4, 1] should be an element of 3-Bounded Partitions """
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