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

class sage.combinat.ncsf_qsym.generic_basis_code.AlgebraMorphism(domain, on_generators, position=0, codomain=None, category=None, anti=False)

Bases: sage.modules.with_basis.morphism.ModuleMorphismByLinearity

A class for algebra morphism defined on a free algebra from the image of the generators

class sage.combinat.ncsf_qsym.generic_basis_code.BasesOfQSymOrNCSF(parent_with_realization)

Bases: sage.categories.realizations.Category_realization_of_parent

class ElementMethods

Bases: object

degree()

The maximum of the degrees of the homogeneous summands.

See also

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
degree_negation()

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]

Todo

Generalize this to all graded vector spaces?

duality_pairing(y)

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
skew_by(y, side='left')

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
class ParentMethods

Bases: object

alternating_sum_of_compositions(n)

Alternating sum over compositions of n.

Note that this differs from the method 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]
alternating_sum_of_fatter_compositions(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]
alternating_sum_of_finer_compositions(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]
counit_on_basis(I)

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
degree_negation(element)

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]

Todo

Generalize this to all graded vector spaces?

degree_on_basis(I)

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
duality_pairing(x, y)

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
duality_pairing_by_coercion(x, y)

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
duality_pairing_matrix(basis, degree)

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]
one_basis()

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()
[]
skew(x, y, side='left')

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
sum_of_fatter_compositions(composition)

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]
sum_of_finer_compositions(composition)

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]
sum_of_partition_rearrangements(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[]
super_categories()
class sage.combinat.ncsf_qsym.generic_basis_code.GradedModulesWithInternalProduct(base, name=None)

Bases: sage.categories.category_types.Category_over_base_ring

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
class ElementMethods

Bases: object

internal_product(other)

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

\[\langle f * g, h \rangle = \sum_i \left\langle f, h^{\prime}_i \right\rangle \left\langle g, h^{\prime\prime}_i \right\rangle,\]

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

\[S^I * S^J = \sum S^{\operatorname*{comp}M},\]

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 OrderedSetPartition() for the meaning of “ordered set partition”.)

Aliases for internal_product() are itensor() and 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)  # not tested, too long
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(other)

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

\[\langle f * g, h \rangle = \sum_i \left\langle f, h^{\prime}_i \right\rangle \left\langle g, h^{\prime\prime}_i \right\rangle,\]

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

\[S^I * S^J = \sum S^{\operatorname*{comp}M},\]

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 OrderedSetPartition() for the meaning of “ordered set partition”.)

Aliases for internal_product() are itensor() and 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)  # not tested, too long
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
kronecker_product(other)

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

\[\langle f * g, h \rangle = \sum_i \left\langle f, h^{\prime}_i \right\rangle \left\langle g, h^{\prime\prime}_i \right\rangle,\]

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

\[S^I * S^J = \sum S^{\operatorname*{comp}M},\]

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 OrderedSetPartition() for the meaning of “ordered set partition”.)

Aliases for internal_product() are itensor() and 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)  # not tested, too long
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
class ParentMethods

Bases: object

internal_product()

The bilinear product inherited from the isomorphism with the descent algebra.

This is constructed by extending the method internal_product_on_basis() bilinearly, if available, or using the method 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
internal_product_on_basis(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
itensor()

The bilinear product inherited from the isomorphism with the descent algebra.

This is constructed by extending the method internal_product_on_basis() bilinearly, if available, or using the method 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
kronecker_product()

The bilinear product inherited from the isomorphism with the descent algebra.

This is constructed by extending the method internal_product_on_basis() bilinearly, if available, or using the method 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
class Realizations(category, *args)

Bases: sage.categories.realizations.RealizationsCategory

class ParentMethods

Bases: object

internal_product_by_coercion(left, right)

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
super_categories()

EXAMPLES:

sage: from sage.combinat.ncsf_qsym.generic_basis_code import GradedModulesWithInternalProduct
sage: GradedModulesWithInternalProduct(ZZ).super_categories()
[Category of graded modules over Integer Ring]