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""" Generic code for bases
This is a collection of code that is shared by bases of noncommutative symmetric functions and quasisymmetric functions.
AUTHORS:
- Jason Bandlow - Franco Saliola - Chris Berg """ #***************************************************************************** # Copyright (C) 2010 Jason Bandlow <jbandlow@gmail.com>, # 2012 Franco Saliola <saliola@gmail.com>, # # 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/ #*****************************************************************************
from sage.misc.cachefunc import cached_method from sage.categories.realizations import Category_realization_of_parent from sage.categories.modules_with_basis import ModulesWithBasis from sage.modules.with_basis.morphism import ModuleMorphismByLinearity from sage.combinat.composition import Compositions, Composition from sage.combinat.partition import Partition from sage.combinat.permutation import Permutations from sage.rings.integer import Integer from sage.categories.all import AlgebrasWithBasis from sage.misc.lazy_attribute import lazy_attribute from sage.misc.abstract_method import abstract_method from sage.categories.category_types import Category_over_base_ring from sage.categories.realizations import RealizationsCategory
class BasesOfQSymOrNCSF(Category_realization_of_parent):
def _repr_object_names(self): r""" Return the name of the objects of this category.
TESTS::
sage: from sage.combinat.ncsf_qsym.generic_basis_code import BasesOfQSymOrNCSF sage: QSym = QuasiSymmetricFunctions(QQ) sage: C = BasesOfQSymOrNCSF(QSym) sage: C._repr_object_names() 'bases of Non-Commutative Symmetric Functions or Quasisymmetric functions over the Rational Field' sage: C Category of bases of Non-Commutative Symmetric Functions or Quasisymmetric functions over the Rational Field
"""
def super_categories(self): r""" TESTS::
sage: from sage.combinat.ncsf_qsym.generic_basis_code import BasesOfQSymOrNCSF sage: QSym = QuasiSymmetricFunctions(QQ) sage: BasesOfQSymOrNCSF(QSym).super_categories() [Category of realizations of Quasisymmetric functions over the Rational Field, Category of graded hopf algebras with basis over Rational Field, Join of Category of realizations of hopf algebras over Rational Field and Category of graded algebras over Rational Field] """ GradedHopfAlgebrasWithBasis(R), GradedHopfAlgebras(R).Realizations()]
class ParentMethods:
def _repr_(self): """ TESTS::
sage: S = NonCommutativeSymmetricFunctions(QQ).complete() sage: S._repr_() 'Non-Commutative Symmetric Functions over the Rational Field in the Complete basis' sage: F = QuasiSymmetricFunctions(ZZ).Fundamental() sage: F._repr_() 'Quasisymmetric functions over the Integer Ring in the Fundamental basis' """
def __getitem__(self, c, *rest): """ This method implements the abuses of notations::
sage: Psi = NonCommutativeSymmetricFunctions(QQ).Psi() sage: Psi[2,1] Psi[2, 1] sage: Psi[[2,1]] Psi[2, 1] sage: Psi[Composition([2,1])] Psi[2, 1]
.. todo::
This should call ``super.monomial`` if the input can't be made into a composition so as not to interfere with the standard notation ``Psi['x,y,z']``.
This could possibly be shared with Sym, FQSym, and other algebras with bases indexed by list-like objects """ else: else:
# could go to Algebras(...).Graded().Connected() or Modules(...).Graded().Connected() @cached_method def one_basis(self): r""" Return the empty composition.
OUTPUT:
- The empty composition.
EXAMPLES::
sage: L=NonCommutativeSymmetricFunctions(QQ).L() sage: parent(L) <class 'sage.combinat.ncsf_qsym.ncsf.NonCommutativeSymmetricFunctions.Elementary_with_category'> sage: parent(L).one_basis() [] """
# Combinatorial rules
def sum_of_finer_compositions(self, composition): r""" Return the sum of all finer compositions.
INPUT:
- ``composition`` -- a composition
OUTPUT:
- The sum of all basis ``self`` elements which are indexed by compositions finer than ``composition``.
EXAMPLES::
sage: L=NonCommutativeSymmetricFunctions(QQ).L() sage: L.sum_of_finer_compositions(Composition([2,1])) L[1, 1, 1] + L[2, 1] sage: R=NonCommutativeSymmetricFunctions(QQ).R() sage: R.sum_of_finer_compositions(Composition([1,3])) R[1, 1, 1, 1] + R[1, 1, 2] + R[1, 2, 1] + R[1, 3] """
def sum_of_fatter_compositions(self, composition): r""" Return the sum of all fatter compositions.
INPUT:
- ``composition`` -- a composition
OUTPUT:
- the sum of all basis elements which are indexed by compositions fatter (coarser?) than ``composition``.
EXAMPLES::
sage: L=NonCommutativeSymmetricFunctions(QQ).L() sage: L.sum_of_fatter_compositions(Composition([2,1])) L[2, 1] + L[3] sage: R=NonCommutativeSymmetricFunctions(QQ).R() sage: R.sum_of_fatter_compositions(Composition([1,3])) R[1, 3] + R[4] """
def alternating_sum_of_compositions(self, n): r""" Alternating sum over compositions of ``n``.
Note that this differs from the method :meth:`alternating_sum_of_finer_compositions` because the coefficient of the composition `1^n` is positive. This method is used in the expansion of the elementary generators into the complete generators and vice versa.
INPUT:
- ``n`` -- a positive integer
OUTPUT:
- The expansion of the complete generator indexed by ``n`` into the elementary basis.
EXAMPLES::
sage: L=NonCommutativeSymmetricFunctions(QQ).L() sage: L.alternating_sum_of_compositions(0) L[] sage: L.alternating_sum_of_compositions(1) L[1] sage: L.alternating_sum_of_compositions(2) L[1, 1] - L[2] sage: L.alternating_sum_of_compositions(3) L[1, 1, 1] - L[1, 2] - L[2, 1] + L[3] sage: S=NonCommutativeSymmetricFunctions(QQ).S() sage: S.alternating_sum_of_compositions(3) S[1, 1, 1] - S[1, 2] - S[2, 1] + S[3] """ (compo, ring((-1)**(len(compo)))) for compo in Compositions(n) )
def alternating_sum_of_finer_compositions(self, composition, conjugate = False): """ Return the alternating sum of finer compositions in a basis of the non-commutative symmetric functions.
INPUT:
- ``composition`` -- a composition - ``conjugate`` -- (default: ``False``) a boolean
OUTPUT:
- The alternating sum of the compositions finer than ``composition``, in the basis ``self``. The alternation is upon the length of the compositions, and is normalized so that ``composition`` has coefficient `1`. If the variable ``conjugate`` is set to ``True``, then the conjugate of ``composition`` is used instead of ``composition``.
EXAMPLES::
sage: NCSF = NonCommutativeSymmetricFunctions(QQ) sage: elementary = NCSF.elementary() sage: elementary.alternating_sum_of_finer_compositions(Composition([2,2,1])) L[1, 1, 1, 1, 1] - L[1, 1, 2, 1] - L[2, 1, 1, 1] + L[2, 2, 1] sage: elementary.alternating_sum_of_finer_compositions(Composition([1,2])) -L[1, 1, 1] + L[1, 2]
TESTS::
sage: complete = NonCommutativeSymmetricFunctions(ZZ).complete() sage: I = Composition([2]) sage: x = complete.alternating_sum_of_finer_compositions(I) sage: [c.parent() for c in x.coefficients()] [Integer Ring, Integer Ring] """ composition = composition.conjugate()
def alternating_sum_of_fatter_compositions(self, composition): """ Return the alternating sum of fatter compositions in a basis of the non-commutative symmetric functions.
INPUT:
- ``composition`` -- a composition
OUTPUT:
- The alternating sum of the compositions fatter than ``composition``, in the basis ``self``. The alternation is upon the length of the compositions, and is normalized so that ``composition`` has coefficient `1`.
EXAMPLES::
sage: NCSF=NonCommutativeSymmetricFunctions(QQ) sage: elementary = NCSF.elementary() sage: elementary.alternating_sum_of_fatter_compositions(Composition([2,2,1])) L[2, 2, 1] - L[2, 3] - L[4, 1] + L[5] sage: elementary.alternating_sum_of_fatter_compositions(Composition([1,2])) L[1, 2] - L[3]
TESTS::
sage: complete = NonCommutativeSymmetricFunctions(ZZ).complete() sage: I = Composition([1,1]) sage: x = complete.alternating_sum_of_fatter_compositions(I) sage: [c.parent() for c in x.coefficients()] [Integer Ring, Integer Ring] """
def sum_of_partition_rearrangements(self, par): """ Return the sum of all basis elements indexed by compositions which can be sorted to obtain a given partition.
INPUT:
- ``par`` -- a partition
OUTPUT:
- The sum of all ``self`` basis elements indexed by compositions which are permutations of ``par`` (without multiplicity).
EXAMPLES::
sage: NCSF=NonCommutativeSymmetricFunctions(QQ) sage: elementary = NCSF.elementary() sage: elementary.sum_of_partition_rearrangements(Partition([2,2,1])) L[1, 2, 2] + L[2, 1, 2] + L[2, 2, 1] sage: elementary.sum_of_partition_rearrangements(Partition([3,2,1])) L[1, 2, 3] + L[1, 3, 2] + L[2, 1, 3] + L[2, 3, 1] + L[3, 1, 2] + L[3, 2, 1] sage: elementary.sum_of_partition_rearrangements(Partition([])) L[] """
def _comp_to_par(self, comp): """ Return the partition if the composition is actually a partition. Otherwise returns nothing.
INPUT:
- ``comp`` -- a composition
OUTPUT:
- ``comp`` as a partition, if it is sorted; otherwise returns ``None`` (nothing).
EXAMPLES::
sage: NCSF=NonCommutativeSymmetricFunctions(QQ) sage: L = NCSF.elementary() sage: L._comp_to_par(Composition([1,1,3,1,2])) sage: L.sum_of_partition_rearrangements(Composition([])) L[] sage: L._comp_to_par(Composition([3,2,1,1])) [3, 2, 1, 1] """
def degree_on_basis(self, I): r""" Return the degree of the basis element indexed by `I`.
INPUT:
- ``I`` -- a composition
OUTPUT:
- The degree of the non-commutative symmetric function basis element of ``self`` indexed by ``I``. By definition, this is the size of the composition ``I``.
EXAMPLES::
sage: R = NonCommutativeSymmetricFunctions(QQ).ribbon() sage: R.degree_on_basis(Composition([2,3])) 5 sage: M = QuasiSymmetricFunctions(QQ).Monomial() sage: M.degree_on_basis(Composition([3,2])) 5 sage: M.degree_on_basis(Composition([])) 0 """
def skew(self, x, y, side='left'): r""" Return a function ``x`` in ``self`` skewed by a function ``y`` in the Hopf dual of ``self``.
INPUT:
- ``x`` -- a non-commutative or quasi-symmetric function; it is an element of ``self`` - ``y`` -- a quasi-symmetric or non-commutative symmetric function; it is an element of the dual algebra of ``self`` - ``side`` -- (default: ``'left'``) either ``'left'`` or ``'right'``
OUTPUT:
- The result of skewing the element ``x`` by the Hopf algebra element ``y`` (either from the left or from the right, as determined by ``side``), written in the basis ``self``.
EXAMPLES::
sage: S = NonCommutativeSymmetricFunctions(QQ).complete() sage: F = QuasiSymmetricFunctions(QQ).Fundamental() sage: S.skew(S[2,2,2], F[1,1]) S[1, 1, 2] + S[1, 2, 1] + S[2, 1, 1] sage: S.skew(S[2,2,2], F[2]) S[1, 1, 2] + S[1, 2, 1] + S[2, 1, 1] + 3*S[2, 2]
::
sage: R = NonCommutativeSymmetricFunctions(QQ).ribbon() sage: F = QuasiSymmetricFunctions(QQ).Fundamental() sage: R.skew(R[2,2,2], F[1,1]) R[1, 1, 2] + R[1, 2, 1] + R[1, 3] + R[2, 1, 1] + 2*R[2, 2] + R[3, 1] + R[4] sage: R.skew(R[2,2,2], F[2]) R[1, 1, 2] + R[1, 2, 1] + R[1, 3] + R[2, 1, 1] + 3*R[2, 2] + R[3, 1] + R[4]
::
sage: S = NonCommutativeSymmetricFunctions(QQ).S() sage: R = NonCommutativeSymmetricFunctions(QQ).R() sage: M = QuasiSymmetricFunctions(QQ).M() sage: M.skew(M[3,2], S[2]) 0 sage: M.skew(M[3,2], S[2], side='right') M[3] sage: M.skew(M[3,2], S[3]) M[2] sage: M.skew(M[3,2], S[3], side='right') 0
TESTS::
sage: R = NonCommutativeSymmetricFunctions(QQ).R() sage: R.skew([2,1], [1]) Traceback (most recent call last): ... AssertionError: x must be an element of Non-Commutative Symmetric Functions over the Rational Field sage: R([2,1]).skew_by([1]) Traceback (most recent call last): ... AssertionError: y must be an element of Quasisymmetric functions over the Rational Field sage: F = QuasiSymmetricFunctions(QQ).F() sage: F([2,1]).skew_by([1]) Traceback (most recent call last): ... AssertionError: y must be an element of Non-Commutative Symmetric Functions over the Rational Field """ for (IJ, coeff) in x.coproduct() if IJ[1-v] in y) else: return self._skew_by_coercion(x, y, side=side)
def _skew_by_coercion(self, x, y, side='left'): r""" Return a function ``x`` in ``self`` skewed by a function ``y`` in the Hopf dual of ``self`` using coercion.
INPUT:
- ``x`` -- a non-commutative or quasi-symmetric function; it is an element of ``self`` - ``y`` -- a quasi-symmetric or non-commutative symmetric function; it is an element of the dual algebra of ``self`` - ``side`` -- (default: ``'left'``) either ``'left'`` or ``'right'``
OUTPUT:
- The result of skewing the element ``x`` by the Hopf algebra element ``y`` (either from the left or from the right, as determined by ``side``), written in the basis ``self``. This uses coercion to a concrete realization (either the complete basis of non-commutative symmetric functions or the monomial basis of the quasi-symmetric functions).
EXAMPLES::
sage: N = NonCommutativeSymmetricFunctions(QQ) sage: R = NonCommutativeSymmetricFunctions(QQ).R() sage: M = QuasiSymmetricFunctions(QQ).M() sage: M._skew_by_coercion(M[1,2,1,3], R[1]) M[2, 1, 3] sage: M._skew_by_coercion(M[1,2,1,3], R[1],side='right') 0 """
def duality_pairing(self, x, y): r""" The duality pairing between elements of `NSym` and elements of `QSym`.
This is a default implementation that uses ``self.realizations_of().a_realization()`` and its dual basis.
INPUT:
- ``x`` -- an element of ``self`` - ``y`` -- an element in the dual basis of ``self``
OUTPUT:
- The result of pairing the function ``x`` from ``self`` with the function ``y`` from the dual basis of ``self``
EXAMPLES::
sage: R = NonCommutativeSymmetricFunctions(QQ).Ribbon() sage: F = QuasiSymmetricFunctions(QQ).Fundamental() sage: R.duality_pairing(R[1,1,2], F[1,1,2]) 1 sage: R.duality_pairing(R[1,2,1], F[1,1,2]) 0 sage: F.duality_pairing(F[1,2,1], R[1,1,2]) 0
::
sage: S = NonCommutativeSymmetricFunctions(QQ).Complete() sage: M = QuasiSymmetricFunctions(QQ).Monomial() sage: S.duality_pairing(S[1,1,2], M[1,1,2]) 1 sage: S.duality_pairing(S[1,2,1], M[1,1,2]) 0 sage: M.duality_pairing(M[1,1,2], S[1,1,2]) 1 sage: M.duality_pairing(M[1,2,1], S[1,1,2]) 0
::
sage: S = NonCommutativeSymmetricFunctions(QQ).Complete() sage: F = QuasiSymmetricFunctions(QQ).Fundamental() sage: S.duality_pairing(S[1,2], F[1,1,1]) 0 sage: S.duality_pairing(S[1,1,1,1], F[4]) 1
TESTS:
The result has the right parent even if the sum is empty::
sage: x = S.duality_pairing(S.zero(), F.zero()); x 0 sage: parent(x) Rational Field """ else:
def duality_pairing_by_coercion(self, x, y): r""" The duality pairing between elements of NSym and elements of QSym.
This is a default implementation that uses ``self.realizations_of().a_realization()`` and its dual basis.
INPUT:
- ``x`` -- an element of ``self`` - ``y`` -- an element in the dual basis of ``self``
OUTPUT:
- The result of pairing the function ``x`` from ``self`` with the function ``y`` from the dual basis of ``self``
EXAMPLES::
sage: L = NonCommutativeSymmetricFunctions(QQ).Elementary() sage: F = QuasiSymmetricFunctions(QQ).Fundamental() sage: L.duality_pairing_by_coercion(L[1,2], F[1,2]) 0 sage: F.duality_pairing_by_coercion(F[1,2], L[1,2]) 0 sage: L.duality_pairing_by_coercion(L[1,1,1], F[1,2]) 1 sage: F.duality_pairing_by_coercion(F[1,2], L[1,1,1]) 1
TESTS:
The result has the right parent even if the sum is empty::
sage: x = F.duality_pairing_by_coercion(F.zero(), L.zero()); x 0 sage: parent(x) Rational Field """
def duality_pairing_matrix(self, basis, degree): r""" The matrix of scalar products between elements of NSym and elements of QSym.
INPUT:
- ``basis`` -- A basis of the dual Hopf algebra - ``degree`` -- a non-negative integer
OUTPUT:
- The matrix of scalar products between the basis ``self`` and the basis ``basis`` in the dual Hopf algebra in degree ``degree``.
EXAMPLES:
The ribbon basis of NCSF is dual to the fundamental basis of QSym::
sage: R = NonCommutativeSymmetricFunctions(QQ).ribbon() sage: F = QuasiSymmetricFunctions(QQ).Fundamental() sage: R.duality_pairing_matrix(F, 3) [1 0 0 0] [0 1 0 0] [0 0 1 0] [0 0 0 1] sage: F.duality_pairing_matrix(R, 3) [1 0 0 0] [0 1 0 0] [0 0 1 0] [0 0 0 1]
The complete basis of NCSF is dual to the monomial basis of QSym::
sage: S = NonCommutativeSymmetricFunctions(QQ).complete() sage: M = QuasiSymmetricFunctions(QQ).Monomial() sage: S.duality_pairing_matrix(M, 3) [1 0 0 0] [0 1 0 0] [0 0 1 0] [0 0 0 1] sage: M.duality_pairing_matrix(S, 3) [1 0 0 0] [0 1 0 0] [0 0 1 0] [0 0 0 1]
The matrix between the ribbon basis of NCSF and the monomial basis of QSym::
sage: R = NonCommutativeSymmetricFunctions(QQ).ribbon() sage: M = QuasiSymmetricFunctions(QQ).Monomial() sage: R.duality_pairing_matrix(M, 3) [ 1 -1 -1 1] [ 0 1 0 -1] [ 0 0 1 -1] [ 0 0 0 1] sage: M.duality_pairing_matrix(R, 3) [ 1 0 0 0] [-1 1 0 0] [-1 0 1 0] [ 1 -1 -1 1]
The matrix between the complete basis of NCSF and the fundamental basis of QSym::
sage: S = NonCommutativeSymmetricFunctions(QQ).complete() sage: F = QuasiSymmetricFunctions(QQ).Fundamental() sage: S.duality_pairing_matrix(F, 3) [1 1 1 1] [0 1 0 1] [0 0 1 1] [0 0 0 1]
A base case test::
sage: R.duality_pairing_matrix(M,0) [1] """ # TODO: generalize to keys indexing the basis of the graded component [[self.duality_pairing(self[I], basis[J]) \ for J in Compositions(degree)] \ for I in Compositions(degree)])
def counit_on_basis(self, I): r""" The counit is defined by sending all elements of positive degree to zero.
EXAMPLES::
sage: S = NonCommutativeSymmetricFunctions(QQ).S() sage: S.counit_on_basis([1,3]) 0 sage: M = QuasiSymmetricFunctions(QQ).M() sage: M.counit_on_basis([1,3]) 0
TESTS::
sage: S.counit_on_basis([]) 1 sage: S.counit_on_basis(Composition([])) 1 sage: M.counit_on_basis([]) 1 sage: M.counit_on_basis(Composition([])) 1 """ else:
def degree_negation(self, element): r""" Return the image of ``element`` under the degree negation automorphism of ``self``.
The degree negation is the automorphism which scales every homogeneous element of degree `k` by `(-1)^k` (for all `k`).
INPUT:
- ``element`` -- element of ``self``
EXAMPLES::
sage: NSym = NonCommutativeSymmetricFunctions(ZZ) sage: S = NSym.S() sage: f = 2*S[2,1] + 4*S[1,1] - 5*S[1,2] - 3*S[[]] sage: S.degree_negation(f) -3*S[] + 4*S[1, 1] + 5*S[1, 2] - 2*S[2, 1]
sage: QSym = QuasiSymmetricFunctions(QQ) sage: dI = QSym.dualImmaculate() sage: f = -3*dI[2,1] + 4*dI[2] + 2*dI[1] sage: dI.degree_negation(f) -2*dI[1] + 4*dI[2] + 3*dI[2, 1]
TESTS:
Using :meth:`degree_negation` on an element of a different basis works correctly::
sage: NSym = NonCommutativeSymmetricFunctions(QQ) sage: S = NSym.S() sage: Phi = NSym.Phi() sage: S.degree_negation(Phi[2]) -S[1, 1] + 2*S[2] sage: S.degree_negation(Phi[3]) -S[1, 1, 1] + 3/2*S[1, 2] + 3/2*S[2, 1] - 3*S[3] sage: Phi.degree_negation(S[3]) -1/6*Phi[1, 1, 1] - 1/4*Phi[1, 2] - 1/4*Phi[2, 1] - 1/3*Phi[3]
The zero element behaves well::
sage: a = Phi.degree_negation(S.zero()); a 0 sage: parent(a) Non-Commutative Symmetric Functions over the Rational Field in the Phi basis
.. TODO::
Generalize this to all graded vector spaces? """ for lam, a in self(element) ], distinct=True)
class ElementMethods:
def degree_negation(self): r""" Return the image of ``self`` under the degree negation automorphism of the parent of ``self``.
The degree negation is the automorphism which scales every homogeneous element of degree `k` by `(-1)^k` (for all `k`).
Calling ``degree_negation(self)`` is equivalent to calling ``self.parent().degree_negation(self)``.
EXAMPLES::
sage: NSym = NonCommutativeSymmetricFunctions(ZZ) sage: S = NSym.S() sage: f = 2*S[2,1] + 4*S[1,1] - 5*S[1,2] - 3*S[[]] sage: f.degree_negation() -3*S[] + 4*S[1, 1] + 5*S[1, 2] - 2*S[2, 1]
sage: QSym = QuasiSymmetricFunctions(QQ) sage: dI = QSym.dualImmaculate() sage: f = -3*dI[2,1] + 4*dI[2] + 2*dI[1] sage: f.degree_negation() -2*dI[1] + 4*dI[2] + 3*dI[2, 1]
TESTS:
The zero element behaves well::
sage: a = S.zero().degree_negation(); a 0 sage: parent(a) Non-Commutative Symmetric Functions over the Integer Ring in the Complete basis
.. TODO::
Generalize this to all graded vector spaces? """ for lam, a in self ], distinct=True)
def duality_pairing(self, y): r""" The duality pairing between elements of `NSym` and elements of `QSym`.
The complete basis is dual to the monomial basis with respect to this pairing.
INPUT:
- ``y`` -- an element of the dual Hopf algebra of ``self``
OUTPUT:
- The result of pairing ``self`` with ``y``.
EXAMPLES::
sage: R = NonCommutativeSymmetricFunctions(QQ).Ribbon() sage: F = QuasiSymmetricFunctions(QQ).Fundamental() sage: R[1,1,2].duality_pairing(F[1,1,2]) 1 sage: R[1,2,1].duality_pairing(F[1,1,2]) 0
::
sage: L = NonCommutativeSymmetricFunctions(QQ).Elementary() sage: F = QuasiSymmetricFunctions(QQ).Fundamental() sage: L[1,2].duality_pairing(F[1,2]) 0 sage: L[1,1,1].duality_pairing(F[1,2]) 1
"""
def skew_by(self, y, side='left'): r""" The operation which is dual to multiplication by ``y``, where ``y`` is an element of the dual space of ``self``.
This is calculated through the coproduct of ``self`` and the expansion of ``y`` in the dual basis.
INPUT:
- ``y`` -- an element of the dual Hopf algebra of ``self`` - ``side`` -- (Default='left') Either 'left' or 'right'
OUTPUT:
- The result of skewing ``self`` by ``y``, on the side ``side``
EXAMPLES:
Skewing an element of NCSF by an element of QSym::
sage: R = NonCommutativeSymmetricFunctions(QQ).ribbon() sage: F = QuasiSymmetricFunctions(QQ).Fundamental() sage: R([2,2,2]).skew_by(F[1,1]) R[1, 1, 2] + R[1, 2, 1] + R[1, 3] + R[2, 1, 1] + 2*R[2, 2] + R[3, 1] + R[4] sage: R([2,2,2]).skew_by(F[2]) R[1, 1, 2] + R[1, 2, 1] + R[1, 3] + R[2, 1, 1] + 3*R[2, 2] + R[3, 1] + R[4]
Skewing an element of QSym by an element of NCSF::
sage: S = NonCommutativeSymmetricFunctions(QQ).S() sage: R = NonCommutativeSymmetricFunctions(QQ).R() sage: F = QuasiSymmetricFunctions(QQ).F() sage: F[3,2].skew_by(R[1,1]) 0 sage: F[3,2].skew_by(R[1,1], side='right') 0 sage: F[3,2].skew_by(S[1,1,1], side='right') F[2] sage: F[3,2].skew_by(S[1,2], side='right') F[2] sage: F[3,2].skew_by(S[2,1], side='right') 0 sage: F[3,2].skew_by(S[1,1,1]) F[2] sage: F[3,2].skew_by(S[1,1]) F[1, 2] sage: F[3,2].skew_by(S[1]) F[2, 2]
::
sage: S = NonCommutativeSymmetricFunctions(QQ).S() sage: R = NonCommutativeSymmetricFunctions(QQ).R() sage: M = QuasiSymmetricFunctions(QQ).M() sage: M[3,2].skew_by(S[2]) 0 sage: M[3,2].skew_by(S[2], side='right') M[3] sage: M[3,2].skew_by(S[3]) M[2] sage: M[3,2].skew_by(S[3], side='right') 0 """
def degree(self): """ The maximum of the degrees of the homogeneous summands.
.. SEEALSO:: :meth:`~sage.categories.graded_algebras_with_basis.GradedAlgebrasWithBasis.ElementMethods.homogeneous_degree`
EXAMPLES::
sage: S = NonCommutativeSymmetricFunctions(QQ).S() sage: (x, y) = (S[2], S[3]) sage: x.degree() 2 sage: (x^3 + 4*y^2).degree() 6 sage: ((1 + x)^3).degree() 6
::
sage: F = QuasiSymmetricFunctions(QQ).F() sage: (x, y) = (F[2], F[3]) sage: x.degree() 2 sage: (x^3 + 4*y^2).degree() 6 sage: ((1 + x)^3).degree() 6
TESTS::
sage: S = NonCommutativeSymmetricFunctions(QQ).S() sage: S.zero().degree() Traceback (most recent call last): ... ValueError: the zero element does not have a well-defined degree sage: F = QuasiSymmetricFunctions(QQ).F() sage: F.zero().degree() Traceback (most recent call last): ... ValueError: the zero element does not have a well-defined degree """
class AlgebraMorphism(ModuleMorphismByLinearity): # Find a better name """ A class for algebra morphism defined on a free algebra from the image of the generators """ def __init__(self, domain, on_generators, position = 0, codomain = None, category = None, anti = False): """ Given a map on the multiplicative basis of a free algebra, this method returns the algebra morphism that is the linear extension of its image on generators.
INPUT:
- ``domain`` -- an algebra with a multiplicative basis - ``on_generators`` -- a function defined on the index set of the generators - ``codomain`` -- the codomain - ``position`` -- integer; default is 0 - ``category`` -- a category; defaults to None - ``anti`` -- a boolean; defaults to False
OUTPUT:
- module morphism
EXAMPLES:
We construct explicitly an algebra morphism::
sage: from sage.combinat.ncsf_qsym.generic_basis_code import AlgebraMorphism sage: NCSF = NonCommutativeSymmetricFunctions(QQ) sage: Psi = NCSF.Psi() sage: f = AlgebraMorphism(Psi, attrcall('conjugate'), codomain=Psi) sage: f Generic endomorphism of Non-Commutative Symmetric Functions over the Rational Field in the Psi basis
Usually, however, one constructs algebra morphisms using the ``algebra_morphism`` method for an algebra::
sage: NCSF = NonCommutativeSymmetricFunctions(QQ) sage: Psi = NCSF.Psi() sage: def double(i) : return Psi[i,i] sage: f = Psi.algebra_morphism(double, codomain = Psi) sage: f Generic endomorphism of Non-Commutative Symmetric Functions over the Rational Field in the Psi basis sage: f(2*Psi[[]] + 3 * Psi[1,3,2] + Psi[2,4] ) 2*Psi[] + 3*Psi[1, 1, 3, 3, 2, 2] + Psi[2, 2, 4, 4] sage: f.category() Category of endsets of unital magmas and right modules over Rational Field and left modules over Rational Field
When extra properties about the morphism are known, one can specify the category of which it is a morphism::
sage: def negate(i): return -Psi[i] sage: f = Psi.algebra_morphism(negate, codomain = Psi, category = GradedHopfAlgebrasWithBasis(QQ)) sage: f Generic endomorphism of Non-Commutative Symmetric Functions over the Rational Field in the Psi basis sage: f(2*Psi[[]] + 3 * Psi[1,3,2] + Psi[2,4] ) 2*Psi[] - 3*Psi[1, 3, 2] + Psi[2, 4] sage: f.category() Category of endsets of hopf algebras over Rational Field and graded modules over Rational Field
If ``anti`` is true, this returns an anti-algebra morphism::
sage: f = Psi.algebra_morphism(double, codomain = Psi, anti=True) sage: f Generic endomorphism of Non-Commutative Symmetric Functions over the Rational Field in the Psi basis sage: f(2*Psi[[]] + 3 * Psi[1,3,2] + Psi[2,4] ) 2*Psi[] + 3*Psi[2, 2, 3, 3, 1, 1] + Psi[4, 4, 2, 2] sage: f.category() Category of endsets of modules with basis over Rational Field
TESTS::
sage: Psi = NonCommutativeSymmetricFunctions(QQ).Psi() sage: Phi = NonCommutativeSymmetricFunctions(QQ).Phi() sage: f = Psi.algebra_morphism(Phi.antipode_on_generators, codomain=Phi) sage: f(Psi[1, 2, 2, 1]) Phi[1, 2, 2, 1] sage: f(Psi[3, 1, 2]) -Phi[3, 1, 2] sage: f.__class__ <class 'sage.combinat.ncsf_qsym.generic_basis_code.AlgebraMorphism_with_category'> sage: TestSuite(f).run(skip=['_test_nonzero_equal']) """ else:
def __eq__(self, other): """ Check equality.
EXAMPLES::
sage: Psi = NonCommutativeSymmetricFunctions(QQ).Psi() sage: Phi = NonCommutativeSymmetricFunctions(QQ).Phi() sage: f = Psi.algebra_morphism(Phi.antipode_on_generators, codomain=Phi) sage: g = Psi.algebra_morphism(Phi.antipode_on_generators, codomain=Phi) sage: f == g True sage: f is g False """ and self._zero == other._zero and self._on_generators == other._on_generators and self._position == other._position and self._is_module_with_basis_over_same_base_ring == other._is_module_with_basis_over_same_base_ring)
def __ne__(self, other): """ Check equality.
EXAMPLES::
sage: Psi = NonCommutativeSymmetricFunctions(QQ).Psi() sage: Phi = NonCommutativeSymmetricFunctions(QQ).Phi() sage: f = Psi.algebra_morphism(Phi.antipode_on_generators, codomain=Phi) sage: g = Psi.algebra_morphism(Phi.antipode_on_generators, codomain=Phi) sage: f != g False sage: h = Phi.algebra_morphism(Psi.antipode_on_generators, codomain=Psi) sage: f != h True """
def _on_basis(self, c): r""" Computes the image of this morphism on the basis element indexed by ``c``.
INPUT:
- ``c`` -- an iterable that spits out generators
OUTPUT:
- element of the codomain
EXAMPLES::
sage: from sage.combinat.ncsf_qsym.generic_basis_code import AlgebraMorphism sage: NCSF = NonCommutativeSymmetricFunctions(QQ) sage: Psi = NCSF.Psi() sage: Phi = NCSF.Phi() sage: f = AlgebraMorphism(Psi, lambda i : Phi[i,i], codomain=Phi) sage: f._on_basis([ 3, 2 ]) Phi[3, 3, 2, 2]
"""
class GradedModulesWithInternalProduct(Category_over_base_ring): r""" Constructs the class of modules with internal product. This is used to give an internal product structure to the non-commutative symmetric functions.
EXAMPLES::
sage: from sage.combinat.ncsf_qsym.generic_basis_code import GradedModulesWithInternalProduct sage: N = NonCommutativeSymmetricFunctions(QQ) sage: R = N.ribbon() sage: R in GradedModulesWithInternalProduct(QQ) True """ @cached_method def super_categories(self): """ EXAMPLES::
sage: from sage.combinat.ncsf_qsym.generic_basis_code import GradedModulesWithInternalProduct sage: GradedModulesWithInternalProduct(ZZ).super_categories() [Category of graded modules over Integer Ring] """
class ParentMethods: @abstract_method(optional=True) def internal_product_on_basis(self, I, J): """ The internal product of the two basis elements indexed by ``I`` and ``J`` (optional)
INPUT:
- ``I``, ``J`` -- compositions indexing two elements of the basis of self
Returns the internal product of the corresponding basis elements. If this method is implemented, the internal product is defined from it by linearity.
EXAMPLES::
sage: N = NonCommutativeSymmetricFunctions(QQ) sage: S = N.complete() sage: S.internal_product_on_basis([2,2], [1,2,1]) 2*S[1, 1, 1, 1] + S[1, 1, 2] + S[2, 1, 1] sage: S.internal_product_on_basis([2,2], [2,1]) 0 """
@lazy_attribute def internal_product(self): r""" The bilinear product inherited from the isomorphism with the descent algebra.
This is constructed by extending the method :meth:`internal_product_on_basis` bilinearly, if available, or using the method :meth:`~GradedModulesWithInternalProduct.Realizations.ParentMethods.internal_product_by_coercion`.
OUTPUT:
- The internal product map of the algebra the non-commutative symmetric functions.
EXAMPLES::
sage: N = NonCommutativeSymmetricFunctions(QQ) sage: S = N.complete() sage: S.internal_product Generic endomorphism of Non-Commutative Symmetric Functions over the Rational Field in the Complete basis sage: S.internal_product(S[2,2], S[1,2,1]) 2*S[1, 1, 1, 1] + S[1, 1, 2] + S[2, 1, 1] sage: S.internal_product(S[2,2], S[1,2]) 0
::
sage: N = NonCommutativeSymmetricFunctions(QQ) sage: R = N.ribbon() sage: R.internal_product <bound method ....internal_product_by_coercion ...> sage: R.internal_product_by_coercion(R[1, 1], R[1,1]) R[2] sage: R.internal_product(R[2,2], R[1,2]) 0
""" self.module_morphism(self.internal_product_on_basis, position=0, codomain=self), position=1) else:
itensor = internal_product kronecker_product = internal_product
class ElementMethods: def internal_product(self, other): r""" Return the internal product of two non-commutative symmetric functions.
The internal product on the algebra of non-commutative symmetric functions is adjoint to the internal coproduct on the algebra of quasisymmetric functions with respect to the duality pairing between these two algebras. This means, explicitly, that any two non-commutative symmetric functions `f` and `g` and any quasi-symmetric function `h` satisfy
.. MATH::
\langle f * g, h \rangle = \sum_i \left\langle f, h^{\prime}_i \right\rangle \left\langle g, h^{\prime\prime}_i \right\rangle,
where we write `\Delta^{\times}(h)` as `\sum_i h^{\prime}_i \otimes h^{\prime\prime}_i`. Here, `f * g` denotes the internal product of the non-commutative symmetric functions `f` and `g`.
If `f` and `g` are two homogeneous elements of `NSym` having distinct degrees, then the internal product `f * g` is zero.
Explicit formulas can be given for internal products of elements of the complete and the Psi bases. First, the formula for the Complete basis ([NCSF1]_ Proposition 5.1): If `I` and `J` are two compositions of lengths `p` and `q`, respectively, then the corresponding Complete homogeneous non-commutative symmetric functions `S^I` and `S^J` have internal product
.. MATH::
S^I * S^J = \sum S^{\operatorname*{comp}M},
where the sum ranges over all `p \times q`-matrices `M \in \NN^{p \times q}` (with nonnegative integers as entries) whose row sum vector is `I` (that is, the sum of the entries of the `r`-th row is the `r`-th part of `I` for all `r`) and whose column sum vector is `J` (that is, the sum of all entries of the `s`-th row is the `s`-th part of `J` for all `s`). Here, for any `M \in \NN^{p \times q}`, we denote by `\operatorname*{comp}M` the composition obtained by reading the entries of the matrix `M` in the usual order (row by row, proceeding left to right in each row, traversing the rows from top to bottom).
The formula on the Psi basis ([NCSF2]_ Lemma 3.10) is more complicated. Let `I` and `J` be two compositions of lengths `p` and `q`, respectively, having the same size `|I| = |J|`. We denote by `\Psi^K` the element of the Psi basis corresponding to any composition `K`.
- If `p > q`, then `\Psi^I * \Psi^J` is plainly `0`.
- Assume that `p = q`. Let `\widetilde{\delta}_{I, J}` denote the integer `1` if the compositions `I` and `J` are permutations of each other, and the integer `0` otherwise. For every positive integer `i`, let `m_i` denote the number of parts of `I` equal to `i`. Then, `\Psi^I * \Psi^J` equals `\widetilde{\delta}_{I, J} \prod_{i>0} i^{m_i} m_i! \Psi^I`.
- Now assume that `p < q`. Write the composition `I` as `I = (i_1, i_2, \ldots, i_p)`. For every nonempty composition `K = (k_1, k_2, \ldots, k_s)`, denote by `\Gamma_K` the non-commutative symmetric function `k_1 [\ldots [[\Psi_{k_1}, \Psi_{k_2}], \Psi_{k_3}], \ldots \Psi_{k_s}]`. For any subset `A` of `\{ 1, 2, \ldots, q \}`, let `J_A` be the composition obtained from `J` by removing the `r`-th parts for all `r \notin A` (while keeping the `r`-th parts for all `r \in A` in order). Then, `\Psi^I * \Psi^J` equals the sum of `\Gamma_{J_{K_1}} \Gamma_{J_{K_2}} \cdots \Gamma_{J_{K_p}}` over all ordered set partitions `(K_1, K_2, \ldots, K_p)` of `\{ 1, 2, \ldots, q \}` into `p` parts such that each `1 \leq k \leq p` satisfies `\left\lvert J_{K_k} \right\rvert = i_k`. (See :meth:`~sage.combinat.set_partition_ordered.OrderedSetPartition` for the meaning of "ordered set partition".)
Aliases for :meth:`internal_product()` are :meth:`itensor()` and :meth:`kronecker_product()`.
INPUT:
- ``other`` -- another non-commutative symmetric function
OUTPUT:
- The result of taking the internal product of ``self`` with ``other``.
EXAMPLES::
sage: N = NonCommutativeSymmetricFunctions(QQ) sage: S = N.complete() sage: x = S.an_element(); x 2*S[] + 2*S[1] + 3*S[1, 1] sage: x.internal_product(S[2]) 3*S[1, 1] sage: x.internal_product(S[1]) 2*S[1] sage: S[1,2].internal_product(S[1,2]) S[1, 1, 1] + S[1, 2]
Let us check the duality between the inner product and the inner coproduct in degree `4`::
sage: M = QuasiSymmetricFunctions(FiniteField(29)).M() sage: S = NonCommutativeSymmetricFunctions(FiniteField(29)).S() sage: def tensor_incopr(f, g, h): # computes \sum_i \left< f, h'_i \right> \left< g, h''_i \right> ....: result = h.base_ring().zero() ....: h_parent = h.parent() ....: for partition_pair, coeff in h.internal_coproduct().monomial_coefficients().items(): ....: result += coeff * f.duality_pairing(h_parent[partition_pair[0]]) * g.duality_pairing(h_parent[partition_pair[1]]) ....: return result sage: def testall(n): ....: return all( all( all( tensor_incopr(S[u], S[v], M[w]) == (S[u].itensor(S[v])).duality_pairing(M[w]) ....: for w in Compositions(n) ) ....: for v in Compositions(n) ) ....: for u in Compositions(n) ) sage: testall(2) True sage: testall(3) # long time True sage: testall(4) # long time True
The internal product on the algebra of non-commutative symmetric functions commutes with the canonical commutative projection on the symmetric functions::
sage: S = NonCommutativeSymmetricFunctions(ZZ).S() sage: e = SymmetricFunctions(ZZ).e() sage: def int_pr_of_S_in_e(I, J): ....: return (S[I].internal_product(S[J])).to_symmetric_function() sage: all( all( int_pr_of_S_in_e(I, J) ....: == S[I].to_symmetric_function().internal_product(S[J].to_symmetric_function()) ....: for I in Compositions(3) ) ....: for J in Compositions(3) ) True """
itensor = internal_product kronecker_product = internal_product
class Realizations(RealizationsCategory): class ParentMethods: def internal_product_by_coercion(self, left, right): r""" Internal product of ``left`` and ``right``.
This is a default implementation that computes the internal product in the realization specified by ``self.realization_of().a_realization()``.
INPUT:
- ``left`` -- an element of the non-commutative symmetric functions - ``right`` -- an element of the non-commutative symmetric functions
OUTPUT:
- The internal product of ``left`` and ``right``.
EXAMPLES::
sage: S=NonCommutativeSymmetricFunctions(QQ).S() sage: S.internal_product_by_coercion(S[2,1], S[3]) S[2, 1] sage: S.internal_product_by_coercion(S[2,1], S[4]) 0 """
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