Word Quasi-symmetric functions#
AUTHORS:
Travis Scrimshaw (2018-04-09): initial implementation
Darij Grinberg and Amy Pang (2018-04-12): further bases and methods
- class sage.combinat.chas.wqsym.WQSymBases(base, graded)#
Bases:
Category_realization_of_parent
The category of bases of \(WQSym\).
- class ElementMethods#
Bases:
object
- algebraic_complement()#
Return the image of the element
self
of \(WQSym\) under the algebraic complement involution.If \(u = (u_1, u_2, \ldots, u_n)\) is a packed word that contains the letters \(1, 2, \ldots, k\) and no others, then the complement of \(u\) is defined to be the packed word \(\overline{u} := (k+1 - u_1, k+1 - u_2, \ldots, k+1 - u_n)\).
The algebraic complement involution is defined as the linear map \(WQSym \to WQSym\) that sends each basis element \(\mathbf{M}_u\) of the monomial basis of \(WQSym\) to the basis element \(\mathbf{M}_{\overline{u}}\). This is a graded algebra automorphism and a coalgebra anti-automorphism of \(WQSym\). Denoting by \(\overline{f}\) the image of an element \(f \in WQSym\) under the algebraic complement involution, it can be shown that every packed word \(u\) satisfies
\[\overline{\mathbf{M}_u} = \mathbf{M}_{\overline{u}}, \qquad \overline{X_u} = X_{\overline{u}},\]where standard notations for classical bases of \(WQSym\) are being used (that is, \(\mathbf{M}\) for the monomial basis, and \(X\) for the characteristic basis).
This can be restated in terms of ordered set partitions: For any ordered set partition \(R = (R_1, R_2, \ldots, R_k)\), let \(R^r\) denote the ordered set partition \((R_k, R_{k-1}, \ldots, R_1)\); this is known as the reversal of \(R\). Then,
\[\overline{\mathbf{M}_A} = \mathbf{M}_{A^r}, \qquad \overline{X_A} = X_{A^r}\]for any ordered set partition \(A\).
The formula describing algebraic complements on the Q basis (
WordQuasiSymmetricFunctions.StronglyCoarser
) is more complicated, and requires some definitions. We define a partial order \(\leq\) on the set of all ordered set partitions as follows: \(A \leq B\) if and only if \(A\) is strongly finer than \(B\) (seeis_strongly_finer()
for a definition of this). The length \(\ell(R)\) of an ordered set partition \(R\) shall be defined as the number of parts of \(R\). Use the notation \(Q\) for the Q basis. For any ordered set partition \(A\) of \([n]\), we have\[\overline{Q_A} = \sum_P c_{A, P} Q_P,\]where the sum is over all ordered set partitions \(P\) of \([n]\), and where the coefficient \(c_{A, P}\) is defined as follows:
If there exists an ordered set partition \(R\) satisfying \(R \leq P\) and \(A \leq R^r\), then this \(R\) is unique, and \(c_{A, P} = \left(-1\right)^{\ell(R) - \ell(P)}\).
If there exists no such \(R\), then \(c_{A, P} = 0\).
The formula describing algebraic complements on the \(\Phi\) basis (
WordQuasiSymmetricFunctions.StronglyFiner
) is identical to the above formula for the Q basis, except that the \(\leq\) sign has to be replaced by \(\geq\) in the definition of the coefficients \(c_{A, P}\). In fact, both formulas are particular cases of a general formula for involutions: Assume that \(V\) is an (additive) abelian group, and that \(I\) is a poset. For each \(i \in I\), let \(M_i\) be an element of \(V\). Also, let \(\omega\) be an involution of the set \(I\) (not necessarily order-preserving or order-reversing), and let \(\omega'\) be an involutive group endomorphism of \(V\) such that each \(i \in I\) satisfies \(\omega'(M_i) = M_{\omega(i)}\). For each \(i \in I\), let \(F_i = \sum_{j \geq i} M_j\), where we assume that the sum is finite. Then, each \(i \in I\) satisfies\[\begin{split}\omega'(F_i) = \sum_j \sum_{\substack{k \leq j; \\ \omega(k) \geq i}} \mu(k, j) F_j,\end{split}\]where \(\mu\) denotes the Möbius function. This formula becomes particularly useful when the \(k\) satisfying \(k \leq j\) and \(\omega(k) \geq i\) is unique (if it exists). In our situation, \(V\) is \(WQSym\), and \(I\) is the set of ordered set partitions equipped either with the \(\leq\) partial order defined above or with its opposite order. The \(M_i\) is the \(\mathbf{M}_A\), whereas the \(F_i\) is either \(Q_i\) or \(\Phi_i\).
If we denote the star involution (
star_involution()
) of the quasisymmetric functions by \(f \mapsto f^{\ast}\), and if we let \(\pi\) be the canonical projection \(WQSym \to QSym\), then each \(f \in WQSym\) satisfies \(\pi(\overline{f}) = (\pi(f))^{\ast}\).See also
EXAMPLES:
Recall that the index set for the bases of \(WQSym\) is given by ordered set partitions, not packed words. Translated into the language of ordered set partitions, the algebraic complement involution acts on the Monomial basis by reversing the ordered set partition. In other words, we have
\[\overline{\mathbf{M}_{(P_1, P_2, \ldots, P_k)}} = \mathbf{M}_{(P_k, P_{k-1}, \ldots, P_1)}\]for any standard ordered set partition \((P_1, P_2, \ldots, P_k)\). Let us check this in practice:
sage: WQSym = algebras.WQSym(ZZ) sage: M = WQSym.M() sage: M[[1,3],[2]].algebraic_complement() M[{2}, {1, 3}] sage: M[[1,4],[2,5],[3,6]].algebraic_complement() M[{3, 6}, {2, 5}, {1, 4}] sage: (3*M[[1]] - 4*M[[]] + 5*M[[1],[2]]).algebraic_complement() -4*M[] + 3*M[{1}] + 5*M[{2}, {1}] sage: X = WQSym.X() sage: X[[1,3],[2]].algebraic_complement() X[{2}, {1, 3}] sage: C = WQSym.C() sage: C[[1,3],[2]].algebraic_complement() -C[{1, 2, 3}] - C[{1, 3}, {2}] + C[{2}, {1, 3}] sage: Q = WQSym.Q() sage: Q[[1,2],[5,6],[3,4]].algebraic_complement() Q[{3, 4}, {1, 2, 5, 6}] + Q[{3, 4}, {5, 6}, {1, 2}] - Q[{3, 4, 5, 6}, {1, 2}] sage: Phi = WQSym.Phi() sage: Phi[[2], [1,3]].algebraic_complement() -Phi[{1}, {3}, {2}] + Phi[{1, 3}, {2}] + Phi[{3}, {1}, {2}]
The algebraic complement involution intertwines the antipode and the inverse of the antipode:
sage: all( M(I).antipode().algebraic_complement().antipode() # long time ....: == M(I).algebraic_complement() ....: for I in OrderedSetPartitions(4) ) True
Testing the \(\pi(\overline{f}) = (\pi(f))^{\ast}\) relation:
sage: all( M[I].algebraic_complement().to_quasisymmetric_function() ....: == M[I].to_quasisymmetric_function().star_involution() ....: for I in OrderedSetPartitions(4) ) True
Todo
Check further commutative squares.
- coalgebraic_complement()#
Return the image of the element
self
of \(WQSym\) under the coalgebraic complement involution.If \(u = (u_1, u_2, \ldots, u_n)\) is a packed word, then the reversal of \(u\) is defined to be the packed word \((u_n, u_{n-1}, \ldots, u_1)\). This reversal is denoted by \(u^r\).
The coalgebraic complement involution is defined as the linear map \(WQSym \to WQSym\) that sends each basis element \(\mathbf{M}_u\) of the monomial basis of \(WQSym\) to the basis element \(\mathbf{M}_{u^r}\). This is a graded coalgebra automorphism and an algebra anti-automorphism of \(WQSym\). Denoting by \(f^r\) the image of an element \(f \in WQSym\) under the coalgebraic complement involution, it can be shown that every packed word \(u\) satisfies
\[(\mathbf{M}_u)^r = \mathbf{M}_{u^r}, \qquad (X_u)^r = X_{u^r},\]where standard notations for classical bases of \(WQSym\) are being used (that is, \(\mathbf{M}\) for the monomial basis, and \(X\) for the characteristic basis).
This can be restated in terms of ordered set partitions: For any ordered set partition \(R\) of \([n]\), let \(\overline{R}\) denote the complement of \(R\) (defined in
complement()
). Then,\[(\mathbf{M}_A)^r = \mathbf{M}_{\overline{A}}, \qquad (X_A)^r = X_{\overline{A}}\]for any ordered set partition \(A\).
Recall that \(WQSym\) is a subring of the ring of all bounded-degree noncommutative power series in countably many indeterminates. The latter ring has an obvious continuous algebra anti-endomorphism which sends each letter \(x_i\) to \(x_i\) (and thus sends each monomial \(x_{i_1} x_{i_2} \cdots x_{i_n}\) to \(x_{i_n} x_{i_{n-1}} \cdots x_{i_1}\)). This anti-endomorphism is actually an involution. The coalgebraic complement involution is simply the restriction of this involution to the subring \(WQSym\).
The formula describing coalgebraic complements on the Q basis (
WordQuasiSymmetricFunctions.StronglyCoarser
) is more complicated, and requires some definitions. We define a partial order \(\leq\) on the set of all ordered set partitions as follows: \(A \leq B\) if and only if \(A\) is strongly finer than \(B\) (seeis_strongly_finer()
for a definition of this). The length \(\ell(R)\) of an ordered set partition \(R\) shall be defined as the number of parts of \(R\). Use the notation \(Q\) for the Q basis. For any ordered set partition \(A\) of \([n]\), we have\[(Q_A)^r = \sum_P c_{A, P} Q_P ,\]where the sum is over all ordered set partitions \(P\) of \([n]\), and where the coefficient \(c_{A, P}\) is defined as follows:
If there exists an ordered set partition \(R\) satisfying \(R \leq P\) and \(A \leq \overline{R}\), then this \(R\) is unique, and \(c_{A, P} = \left(-1\right)^{\ell(R) - \ell(P)}\).
If there exists no such \(R\), then \(c_{A, P} = 0\).
The formula describing coalgebraic complements on the \(\Phi\) basis (
WordQuasiSymmetricFunctions.StronglyFiner
) is identical to the above formula for the Q basis, except that the \(\leq\) sign has to be replaced by \(\geq\) in the definition of the coefficients \(c_{A, P}\). In fact, both formulas are particular cases of the general formula for involutions described in the documentation ofalgebraic_complement()
.If we let \(\pi\) be the canonical projection \(WQSym \to QSym\), then each \(f \in WQSym\) satisfies \(\pi(f^r) = \pi(f)\).
See also
EXAMPLES:
Recall that the index set for the bases of \(WQSym\) is given by ordered set partitions, not packed words. Translated into the language of ordered set partitions, the coalgebraic complement involution acts on the Monomial basis by complementing the ordered set partition. In other words, we have
\[(\mathbf{M}_A)^r = \mathbf{M}_{\overline{A}}\]for any standard ordered set partition \(P\). Let us check this in practice:
sage: WQSym = algebras.WQSym(ZZ) sage: M = WQSym.M() sage: M[[1,3],[2]].coalgebraic_complement() M[{1, 3}, {2}] sage: M[[1,2],[3]].coalgebraic_complement() M[{2, 3}, {1}] sage: M[[1], [4], [2,3]].coalgebraic_complement() M[{4}, {1}, {2, 3}] sage: M[[1,4],[2,5],[3,6]].coalgebraic_complement() M[{3, 6}, {2, 5}, {1, 4}] sage: (3*M[[1]] - 4*M[[]] + 5*M[[1],[2]]).coalgebraic_complement() -4*M[] + 3*M[{1}] + 5*M[{2}, {1}] sage: X = WQSym.X() sage: X[[1,3],[2]].coalgebraic_complement() X[{1, 3}, {2}] sage: C = WQSym.C() sage: C[[1,3],[2]].coalgebraic_complement() C[{1, 3}, {2}] sage: Q = WQSym.Q() sage: Q[[1,2],[5,6],[3,4]].coalgebraic_complement() Q[{1, 2, 5, 6}, {3, 4}] + Q[{5, 6}, {1, 2}, {3, 4}] - Q[{5, 6}, {1, 2, 3, 4}] sage: Phi = WQSym.Phi() sage: Phi[[2], [1,3]].coalgebraic_complement() -Phi[{2}, {1}, {3}] + Phi[{2}, {1, 3}] + Phi[{2}, {3}, {1}]
The coalgebraic complement involution intertwines the antipode and the inverse of the antipode:
sage: all( M(I).antipode().coalgebraic_complement().antipode() # long time ....: == M(I).coalgebraic_complement() ....: for I in OrderedSetPartitions(4) ) True
Testing the \(\pi(f^r) = \pi(f)\) relation above:
sage: all( M[I].coalgebraic_complement().to_quasisymmetric_function() ....: == M[I].to_quasisymmetric_function() ....: for I in OrderedSetPartitions(4) ) True
Todo
Check further commutative squares.
- star_involution()#
Return the image of the element
self
of \(WQSym\) under the star involution.The star involution is the composition of the algebraic complement involution (
algebraic_complement()
) with the coalgebraic complement involution (coalgebraic_complement()
). The composition can be performed in either order, as the involutions commute.The star involution is a graded Hopf algebra anti-automorphism of \(WQSym\). Let \(f^{\ast}\) denote the image of an element \(f \in WQSym\) under the star involution. Let \(\mathbf{M}\), \(X\), \(Q\) and \(\Phi\) stand for the monomial, characteristic, Q and Phi bases of \(WQSym\). For any ordered set partition \(A\) of \([n]\), we let \(A^{\ast}\) denote the complement (
complement()
) of the reversal (reversed()
) of \(A\). Then, for any ordered set partition \(A\) of \([n]\), we have\[(\mathbf{M}_A)^{\ast} = \mathbf{M}_{A^{\ast}}, \qquad (X_A)^{\ast} = X_{A^{\ast}}, \qquad (Q_A)^{\ast} = Q_{A^{\ast}}, \qquad (\Phi_A)^{\ast} = \Phi_{A^{\ast}} .\]The star involution (
star_involution()
) on the ring of noncommutative symmetric functions is a restriction of the star involution on \(WQSym\).If we denote the star involution (
star_involution()
) of the quasisymmetric functions by \(f \mapsto f^{\ast}\), and if we let \(\pi\) be the canonical projection \(WQSym \to QSym\), then each \(f \in WQSym\) satisfies \(\pi(f^{\ast}) = (\pi(f))^{\ast}\).Todo
More commutative diagrams? FQSym and FSym need their own star_involution methods defined first.
EXAMPLES:
Keep in mind that the default input method for basis keys of \(WQSym\) is by entering an ordered set partition, not a packed word. Let us check the basis formulas for the star involution:
sage: WQSym = algebras.WQSym(ZZ) sage: M = WQSym.M() sage: M[[1,3], [2,4,5]].star_involution() M[{1, 2, 4}, {3, 5}] sage: M[[1,3],[2]].star_involution() M[{2}, {1, 3}] sage: M[[1,4],[2,5],[3,6]].star_involution() M[{1, 4}, {2, 5}, {3, 6}] sage: (3*M[[1]] - 4*M[[]] + 5*M[[1],[2]]).star_involution() -4*M[] + 3*M[{1}] + 5*M[{1}, {2}] sage: X = WQSym.X() sage: X[[1,3],[2]].star_involution() X[{2}, {1, 3}] sage: C = WQSym.C() sage: C[[1,3],[2]].star_involution() -C[{1, 2, 3}] - C[{1, 3}, {2}] + C[{2}, {1, 3}] sage: Q = WQSym.Q() sage: Q[[1,3], [2,4,5]].star_involution() Q[{1, 2, 4}, {3, 5}] sage: Phi = WQSym.Phi() sage: Phi[[1,3], [2,4,5]].star_involution() Phi[{1, 2, 4}, {3, 5}]
Testing the formulas for \((Q_A)^{\ast}\) and \((\Phi_A)^{\ast}\):
sage: all(Q[A].star_involution() == Q[A.complement().reversed()] for A in OrderedSetPartitions(4)) True sage: all(Phi[A].star_involution() == Phi[A.complement().reversed()] for A in OrderedSetPartitions(4)) True
The star involution commutes with the antipode:
sage: all( M(I).antipode().star_involution() # long time ....: == M(I).star_involution().antipode() ....: for I in OrderedSetPartitions(4) ) True
Testing the \(\pi(f^{\ast}) = (\pi(f))^{\ast}\) relation:
sage: all( M[I].star_involution().to_quasisymmetric_function() ....: == M[I].to_quasisymmetric_function().star_involution() ....: for I in OrderedSetPartitions(4) ) True
Testing the fact that the star involution on the noncommutative symmetric functions is a restriction of the star involution on \(WQSym\):
sage: NCSF = NonCommutativeSymmetricFunctions(QQ) sage: R = NCSF.R() sage: all(R[I].star_involution().to_fqsym().to_wqsym() ....: == R[I].to_fqsym().to_wqsym().star_involution() ....: for I in Compositions(4)) True
Todo
Check further commutative squares.
- to_quasisymmetric_function()#
The projection of
self
to the ring \(QSym\) of quasisymmetric functions.There is a canonical projection \(\pi : WQSym \to QSym\) that sends every element \(\mathbf{M}_P\) of the monomial basis of \(WQSym\) to the monomial quasisymmetric function \(M_c\), where \(c\) is the composition whose parts are the sizes of the blocks of \(P\). This \(\pi\) is a ring homomorphism.
OUTPUT:
an element of the quasisymmetric functions in the monomial basis
EXAMPLES:
sage: M = algebras.WQSym(QQ).M() sage: M[[1,3],[2]].to_quasisymmetric_function() M[2, 1] sage: (M[[1,3],[2]] + 3*M[[2,3],[1]] - M[[1,2,3],]).to_quasisymmetric_function() 4*M[2, 1] - M[3] sage: X, Y = M[[1,3],[2]], M[[1,2,3],] sage: X.to_quasisymmetric_function() * Y.to_quasisymmetric_function() == (X*Y).to_quasisymmetric_function() True sage: C = algebras.WQSym(QQ).C() sage: C[[2,3],[1,4]].to_quasisymmetric_function() == M(C[[2,3],[1,4]]).to_quasisymmetric_function() True sage: C2 = algebras.WQSym(GF(2)).C() sage: C2[[1,2],[3,4]].to_quasisymmetric_function() M[2, 2] sage: C2[[2,3],[1,4]].to_quasisymmetric_function() M[4]
- class ParentMethods#
Bases:
object
- degree_on_basis(t)#
Return the degree of an ordered set partition in the algebra of word quasi-symmetric functions.
This is the sum of the sizes of the blocks of the ordered set partition.
EXAMPLES:
sage: A = algebras.WQSym(QQ).M() sage: u = OrderedSetPartition([[2,1],]) sage: A.degree_on_basis(u) 2 sage: u = OrderedSetPartition([[2], [1]]) sage: A.degree_on_basis(u) 2
- is_commutative()#
Return whether
self
is commutative.EXAMPLES:
sage: M = algebras.WQSym(ZZ).M() sage: M.is_commutative() False
- is_field(proof=True)#
Return whether
self
is a field.EXAMPLES:
sage: M = algebras.WQSym(QQ).M() sage: M.is_field() False
- one_basis()#
Return the index of the unit.
EXAMPLES:
sage: A = algebras.WQSym(QQ).M() sage: A.one_basis() []
- super_categories()#
The super categories of
self
.EXAMPLES:
sage: from sage.combinat.chas.wqsym import WQSymBases sage: WQSym = algebras.WQSym(ZZ) sage: bases = WQSymBases(WQSym, True) sage: bases.super_categories() [Category of realizations of Word Quasi-symmetric functions over Integer Ring, Join of Category of realizations of hopf algebras over Integer Ring and Category of graded algebras over Integer Ring and Category of graded coalgebras over Integer Ring, Category of graded connected hopf algebras with basis over Integer Ring] sage: bases = WQSymBases(WQSym, False) sage: bases.super_categories() [Category of realizations of Word Quasi-symmetric functions over Integer Ring, Join of Category of realizations of hopf algebras over Integer Ring and Category of graded algebras over Integer Ring and Category of graded coalgebras over Integer Ring, Join of Category of filtered connected hopf algebras with basis over Integer Ring and Category of graded algebras over Integer Ring and Category of graded coalgebras over Integer Ring]
- class sage.combinat.chas.wqsym.WQSymBasis_abstract(alg, graded=True)#
Bases:
CombinatorialFreeModule
,BindableClass
Abstract base class for bases of \(WQSym\).
This must define two attributes:
_prefix
– the basis prefix_basis_name
– the name of the basis (must match one of the names that the basis can be constructed from \(WQSym\))
- an_element()#
Return an element of
self
.EXAMPLES:
sage: M = algebras.WQSym(QQ).M() sage: M.an_element() M[{1}] + 2*M[{1}, {2}]
- options = Current options for WordQuasiSymmetricFunctions element - display: normal - objects: compositions#
- some_elements()#
Return some elements of the word quasi-symmetric functions.
EXAMPLES:
sage: M = algebras.WQSym(QQ).M() sage: M.some_elements() [M[], M[{1}], M[{1, 2}], M[{1}] + M[{1}, {2}], M[] + 1/2*M[{1}]]
- class sage.combinat.chas.wqsym.WordQuasiSymmetricFunctions(R)#
Bases:
UniqueRepresentation
,Parent
The word quasi-symmetric functions.
The ring of word quasi-symmetric functions can be defined as a subring of the ring of all bounded-degree noncommutative power series in countably many indeterminates (i.e., elements in \(R \langle \langle x_1, x_2, x_3, \ldots \rangle \rangle\) of bounded degree). Namely, consider words over the alphabet \(\{1, 2, 3, \ldots\}\); every noncommutative power series is an infinite \(R\)-linear combination of these words. For each such word \(w\), we define the packing of \(w\) to be the word \(\operatorname{pack}(w)\) that is obtained from \(w\) by replacing the smallest letter that appears in \(w\) by \(1\), the second-smallest letter that appears in \(w\) by \(2\), etc. (for example, \(\operatorname{pack}(4112774) = 3112443\)). A word \(w\) is said to be packed if \(\operatorname{pack}(w) = w\). For each packed word \(u\), we define the noncommutative power series \(\mathbf{M}_u = \sum w\), where the sum ranges over all words \(w\) satisfying \(\operatorname{pack}(w) = u\). The span of these power series \(\mathbf{M}_u\) is a subring of the ring of all noncommutative power series; it is called the ring of word quasi-symmetric functions, and is denoted by \(WQSym\).
For each nonnegative integer \(n\), there is a bijection between packed words of length \(n\) and ordered set partitions of \(\{1, 2, \ldots, n\}\). Under this bijection, a packed word \(u = (u_1, u_2, \ldots, u_n)\) of length \(n\) corresponds to the ordered set partition \(P = (P_1, P_2, \ldots, P_k)\) of \(\{1, 2, \ldots, n\}\) whose \(i\)-th part \(P_i\) (for each \(i\)) is the set of all \(j \in \{1, 2, \ldots, n\}\) such that \(u_j = i\).
The basis element \(\mathbf{M}_u\) is also denoted as \(\mathbf{M}_P\) in this situation. The basis \((\mathbf{M}_P)_P\) is called the Monomial basis and is implemented as
Monomial
.Other bases are the cone basis (aka C basis), the characteristic basis (aka X basis), the Q basis and the Phi basis.
Bases of \(WQSym\) are implemented (internally) using ordered set partitions. However, the user may access specific basis vectors using either packed words or ordered set partitions. See the examples below, noting especially the section on ambiguities.
\(WQSym\) is endowed with a connected graded Hopf algebra structure (see Section 2.2 of [NoThWi08], Section 1.1 of [FoiMal14] and Section 4.3.2 of [MeNoTh11]) given by
\[\Delta(\mathbf{M}_{(P_1,\ldots,P_{\ell})}) = \sum_{i=0}^{\ell} \mathbf{M}_{\operatorname{st}(P_1, \ldots, P_i)} \otimes \mathbf{M}_{\operatorname{st}(P_{i+1}, \ldots, P_{\ell})}.\]Here, for any ordered set partition \((Q_1, \ldots, Q_k)\) of a finite set \(Z\) of integers, we let \(\operatorname{st}(Q_1, \ldots, Q_k)\) denote the set partition obtained from \(Z\) by replacing the smallest element appearing in it by \(1\), the second-smallest element by \(2\), and so on.
A rule for multiplying elements of the monomial basis relies on the quasi-shuffle product of two ordered set partitions. The quasi-shuffle product \(\Box\) is given by
ShuffleProduct_overlapping
with the+
operation in the overlapping of the shuffles being the union of the sets. The product \(\mathbf{M}_P \mathbf{M}_Q\) for two ordered set partitions \(P\) and \(Q\) of \([n]\) and \([m]\) is then given by\[\mathbf{M}_P \mathbf{M}_Q = \sum_{R \in P \Box Q^+} \mathbf{M}_R ,\]where \(Q^+\) means \(Q\) with all numbers shifted upwards by \(n\).
Sometimes, \(WQSym\) is also denoted as \(NCQSym\).
REFERENCES:
EXAMPLES:
Constructing the algebra and its Monomial basis:
sage: WQSym = algebras.WQSym(ZZ) sage: WQSym Word Quasi-symmetric functions over Integer Ring sage: M = WQSym.M() sage: M Word Quasi-symmetric functions over Integer Ring in the Monomial basis sage: M[[]] M[]
Calling basis elements using packed words:
sage: x = M[1,2,1]; x M[{1, 3}, {2}] sage: x == M[[1,2,1]] == M[Word([1,2,1])] True sage: y = M[1,1,2] - M[1,2,2]; y -M[{1}, {2, 3}] + M[{1, 2}, {3}]
Calling basis elements using ordered set partitions:
sage: z = M[[1,2,3],]; z M[{1, 2, 3}] sage: z == M[[[1,2,3]]] == M[OrderedSetPartition([[1,2,3]])] True sage: M[[1,2],[3]] M[{1, 2}, {3}]
Note that expressions above are output in terms of ordered set partitions, even when input as packed words. Output as packed words can be achieved by modifying the global options. (See
OrderedSetPartitions.options()
for further details.):sage: M.options.objects = "words" sage: y -M[1, 2, 2] + M[1, 1, 2] sage: M.options.display = "compact" sage: y -M[122] + M[112] sage: z M[111]
The options should be reset to display as ordered set partitions:
sage: M.options._reset() sage: z M[{1, 2, 3}]
Illustration of the Hopf algebra structure:
sage: M[[2, 3], [5], [6], [4], [1]].coproduct() M[] # M[{2, 3}, {5}, {6}, {4}, {1}] + M[{1, 2}] # M[{3}, {4}, {2}, {1}] + M[{1, 2}, {3}] # M[{3}, {2}, {1}] + M[{1, 2}, {3}, {4}] # M[{2}, {1}] + M[{1, 2}, {4}, {5}, {3}] # M[{1}] + M[{2, 3}, {5}, {6}, {4}, {1}] # M[] sage: _ == M[5,1,1,4,2,3].coproduct() True sage: M[[1,1,1]] * M[[1,1,2]] # packed words M[{1, 2, 3}, {4, 5}, {6}] + M[{1, 2, 3, 4, 5}, {6}] + M[{4, 5}, {1, 2, 3}, {6}] + M[{4, 5}, {1, 2, 3, 6}] + M[{4, 5}, {6}, {1, 2, 3}] sage: M[[1,2,3],].antipode() # ordered set partition -M[{1, 2, 3}] sage: M[[1], [2], [3]].antipode() -M[{1, 2, 3}] - M[{2, 3}, {1}] - M[{3}, {1, 2}] - M[{3}, {2}, {1}] sage: x = M[[1],[2],[3]] + 3*M[[2],[1]] sage: x.counit() 0 sage: x.antipode() 3*M[{1}, {2}] + 3*M[{1, 2}] - M[{1, 2, 3}] - M[{2, 3}, {1}] - M[{3}, {1, 2}] - M[{3}, {2}, {1}]
Ambiguities
Some ambiguity arises when accessing basis vectors with the dictionary syntax, i.e.,
M[...]
. A common example is when referencing an ordered set partition with one part. For example, in the expressionM[[1,2]]
, does[[1,2]]
refer to an ordered set partition or does[1,2]
refer to a packed word? We choose the latter: if the received arguments do not behave like a tuple of iterables, then view them as describing a packed word. (In the running example, one argument is received, which behaves as a tuple of integers.) Here are a variety of ways to get the same basis vector:sage: x = M[1,1]; x M[{1, 2}] sage: x == M[[1,1]] # treated as word True sage: x == M[[1,2],] == M[[[1,2]]] # treated as ordered set partitions True sage: M[[1,3],[2]] # treat as ordered set partition M[{1, 3}, {2}] sage: M[[1,3],[2]] == M[1,2,1] # treat as word True
Todo
Dendriform structure.
- class Characteristic(alg)#
Bases:
WQSymBasis_abstract
The Characteristic basis of \(WQSym\).
The Characteristic basis is a graded basis \((X_P)\) of \(WQSym\), indexed by ordered set partitions \(P\). It is defined by
\[X_P = (-1)^{\ell(P)} \mathbf{M}_P ,\]where \((\mathbf{M}_P)_P\) denotes the Monomial basis, and where \(\ell(P)\) denotes the number of blocks in an ordered set partition \(P\).
EXAMPLES:
sage: WQSym = algebras.WQSym(QQ) sage: X = WQSym.X(); X Word Quasi-symmetric functions over Rational Field in the Characteristic basis sage: X[[[1,2,3]]] * X[[1,2],[3]] X[{1, 2, 3}, {4, 5}, {6}] - X[{1, 2, 3, 4, 5}, {6}] + X[{4, 5}, {1, 2, 3}, {6}] - X[{4, 5}, {1, 2, 3, 6}] + X[{4, 5}, {6}, {1, 2, 3}] sage: X[[1, 4], [3], [2]].coproduct() X[] # X[{1, 4}, {3}, {2}] + X[{1, 2}] # X[{2}, {1}] + X[{1, 3}, {2}] # X[{1}] + X[{1, 4}, {3}, {2}] # X[] sage: M = WQSym.M() sage: M(X[[1, 2, 3],]) -M[{1, 2, 3}] sage: M(X[[1, 3], [2]]) M[{1, 3}, {2}] sage: X(M[[1, 2, 3],]) -X[{1, 2, 3}] sage: X(M[[1, 3], [2]]) X[{1, 3}, {2}]
- class Element#
Bases:
IndexedFreeModuleElement
- algebraic_complement()#
Return the image of the element
self
of \(WQSym\) under the algebraic complement involution.See
WQSymBases.ElementMethods.algebraic_complement()
for a definition of the involution and for examples.See also
EXAMPLES:
sage: WQSym = algebras.WQSym(ZZ) sage: X = WQSym.X() sage: X[[1,2],[5,6],[3,4]].algebraic_complement() X[{3, 4}, {5, 6}, {1, 2}] sage: X[[3], [1, 2], [4]].algebraic_complement() X[{4}, {1, 2}, {3}]
- coalgebraic_complement()#
Return the image of the element
self
of \(WQSym\) under the coalgebraic complement involution.See
WQSymBases.ElementMethods.coalgebraic_complement()
for a definition of the involution and for examples.See also
EXAMPLES:
sage: WQSym = algebras.WQSym(ZZ) sage: X = WQSym.X() sage: X[[1,2],[5,6],[3,4]].coalgebraic_complement() X[{5, 6}, {1, 2}, {3, 4}] sage: X[[3], [1, 2], [4]].coalgebraic_complement() X[{2}, {3, 4}, {1}]
- star_involution()#
Return the image of the element
self
of \(WQSym\) under the star involution.See
WQSymBases.ElementMethods.star_involution()
for a definition of the involution and for examples.EXAMPLES:
sage: WQSym = algebras.WQSym(ZZ) sage: X = WQSym.X() sage: X[[1,2],[5,6],[3,4]].star_involution() X[{3, 4}, {1, 2}, {5, 6}] sage: X[[3], [1, 2], [4]].star_involution() X[{1}, {3, 4}, {2}]
- class Cone(alg)#
Bases:
WQSymBasis_abstract
The Cone basis of \(WQSym\).
Let \((X_P)_P\) denote the Characteristic basis of \(WQSym\). Denote the quasi-shuffle of two ordered set partitions \(A\) and \(B\) by \(A \Box B\). For an ordered set partition \(P = (P_1, \ldots, P_{\ell})\), we form a list of ordered set partitions \([P] := (P'_1, \ldots, P'_k)\) as follows. Define a strictly decreasing sequence of integers \(\ell + 1 = i_0 > i_1 > \cdots > i_k = 1\) recursively by requiring that \(\min P_{i_j} \leq \min P_a\) for all \(a < i_{j-1}\). Set \(P'_j = (P_{i_j}, \ldots, P_{i_{j-1}-1})\).
The Cone basis \((C_P)_P\) is defined by
\[C_P = \sum_Q X_Q,\]where the sum is over all elements \(Q\) of the quasi-shuffle product \(P'_1 \Box P'_2 \Box \cdots \Box P'_k\) with \([P] = (P'_1, \ldots, P'_k)\).
EXAMPLES:
sage: WQSym = algebras.WQSym(QQ) sage: C = WQSym.C() sage: C Word Quasi-symmetric functions over Rational Field in the Cone basis sage: X = WQSym.X() sage: X(C[[2,3],[1,4]]) X[{1, 2, 3, 4}] + X[{1, 4}, {2, 3}] + X[{2, 3}, {1, 4}] sage: X(C[[1,4],[2,3]]) X[{1, 4}, {2, 3}] sage: X(C[[2,3],[1],[4]]) X[{1}, {2, 3}, {4}] + X[{1}, {2, 3, 4}] + X[{1}, {4}, {2, 3}] + X[{1, 2, 3}, {4}] + X[{2, 3}, {1}, {4}] sage: X(C[[3], [2, 5], [1, 4]]) X[{1, 2, 3, 4, 5}] + X[{1, 2, 4, 5}, {3}] + X[{1, 3, 4}, {2, 5}] + X[{1, 4}, {2, 3, 5}] + X[{1, 4}, {2, 5}, {3}] + X[{1, 4}, {3}, {2, 5}] + X[{2, 3, 5}, {1, 4}] + X[{2, 5}, {1, 3, 4}] + X[{2, 5}, {1, 4}, {3}] + X[{2, 5}, {3}, {1, 4}] + X[{3}, {1, 2, 4, 5}] + X[{3}, {1, 4}, {2, 5}] + X[{3}, {2, 5}, {1, 4}] sage: C(X[[2,3],[1,4]]) -C[{1, 2, 3, 4}] - C[{1, 4}, {2, 3}] + C[{2, 3}, {1, 4}]
REFERENCES:
Section 4 of [Early2017]
Todo
Experiments suggest that
algebraic_complement()
,coalgebraic_complement()
, andstar_involution()
should have reasonable formulas on the C basis; at least the coefficients of the outputs on any element of the C basis seem to be always \(0, 1, -1\). Is this true? What is the formula?- some_elements()#
Return some elements of the word quasi-symmetric functions in the Cone basis.
EXAMPLES:
sage: C = algebras.WQSym(QQ).C() sage: C.some_elements() [C[], C[{1}], C[{1, 2}], C[] + 1/2*C[{1}]]
- class Monomial(alg, graded=True)#
Bases:
WQSymBasis_abstract
The Monomial basis of \(WQSym\).
The family \((\mathbf{M}_u)\), as defined in
WordQuasiSymmetricFunctions
with \(u\) ranging over all packed words, is a basis for the free \(R\)-module \(WQSym\) and called the Monomial basis. Here it is labelled using ordered set partitions.EXAMPLES:
sage: WQSym = algebras.WQSym(QQ) sage: M = WQSym.M(); M Word Quasi-symmetric functions over Rational Field in the Monomial basis sage: sorted(M.basis(2)) [M[{1}, {2}], M[{2}, {1}], M[{1, 2}]]
- coproduct_on_basis(x)#
Return the coproduct of
self
on the basis element indexed by the ordered set partitionx
.EXAMPLES:
sage: M = algebras.WQSym(QQ).M() sage: M.coproduct(M.one()) # indirect doctest M[] # M[] sage: M.coproduct( M([[1]]) ) # indirect doctest M[] # M[{1}] + M[{1}] # M[] sage: M.coproduct( M([[1,2]]) ) M[] # M[{1, 2}] + M[{1, 2}] # M[] sage: M.coproduct( M([[1], [2]]) ) M[] # M[{1}, {2}] + M[{1}] # M[{1}] + M[{1}, {2}] # M[]
- product_on_basis(x, y)#
Return the (associative) \(*\) product of the basis elements of
self
indexed by the ordered set partitions \(x\) and \(y\).This is the shifted quasi-shuffle product of \(x\) and \(y\).
EXAMPLES:
sage: A = algebras.WQSym(QQ).M() sage: x = OrderedSetPartition([[1],[2,3]]) sage: y = OrderedSetPartition([[1,2]]) sage: z = OrderedSetPartition([[1,2],[3]]) sage: A.product_on_basis(x, y) M[{1}, {2, 3}, {4, 5}] + M[{1}, {2, 3, 4, 5}] + M[{1}, {4, 5}, {2, 3}] + M[{1, 4, 5}, {2, 3}] + M[{4, 5}, {1}, {2, 3}] sage: A.product_on_basis(x, z) M[{1}, {2, 3}, {4, 5}, {6}] + M[{1}, {2, 3, 4, 5}, {6}] + M[{1}, {4, 5}, {2, 3}, {6}] + M[{1}, {4, 5}, {2, 3, 6}] + M[{1}, {4, 5}, {6}, {2, 3}] + M[{1, 4, 5}, {2, 3}, {6}] + M[{1, 4, 5}, {2, 3, 6}] + M[{1, 4, 5}, {6}, {2, 3}] + M[{4, 5}, {1}, {2, 3}, {6}] + M[{4, 5}, {1}, {2, 3, 6}] + M[{4, 5}, {1}, {6}, {2, 3}] + M[{4, 5}, {1, 6}, {2, 3}] + M[{4, 5}, {6}, {1}, {2, 3}] sage: A.product_on_basis(y, y) M[{1, 2}, {3, 4}] + M[{1, 2, 3, 4}] + M[{3, 4}, {1, 2}]
- Phi#
alias of
StronglyFiner
- Q#
alias of
StronglyCoarser
- class StronglyCoarser(alg)#
Bases:
WQSymBasis_abstract
The Q basis of \(WQSym\).
We define a partial order \(\leq\) on the set of all ordered set partitions as follows: \(A \leq B\) if and only if \(A\) is strongly finer than \(B\) (see
is_strongly_finer()
for a definition of this).The Q basis \((Q_P)_P\) is a basis of \(WQSym\) indexed by ordered set partitions, and is defined by
\[Q_P = \sum \mathbf{M}_W,\]where the sum is over ordered set partitions \(W\) satisfying \(P \leq W\).
EXAMPLES:
sage: WQSym = algebras.WQSym(QQ) sage: M = WQSym.M(); Q = WQSym.Q() sage: Q Word Quasi-symmetric functions over Rational Field in the Q basis sage: Q(M[[2,3],[1,4]]) Q[{2, 3}, {1, 4}] sage: Q(M[[1,2],[3,4]]) Q[{1, 2}, {3, 4}] - Q[{1, 2, 3, 4}] sage: M(Q[[1,2],[3,4]]) M[{1, 2}, {3, 4}] + M[{1, 2, 3, 4}] sage: M(Q[[2,3],[1],[4]]) M[{2, 3}, {1}, {4}] + M[{2, 3}, {1, 4}] sage: M(Q[[3], [2, 5], [1, 4]]) M[{3}, {2, 5}, {1, 4}] sage: M(Q[[1, 4], [2, 3], [5], [6]]) M[{1, 4}, {2, 3}, {5}, {6}] + M[{1, 4}, {2, 3}, {5, 6}] + M[{1, 4}, {2, 3, 5}, {6}] + M[{1, 4}, {2, 3, 5, 6}] sage: Q[[1, 3], [2]] * Q[[1], [2]] Q[{1, 3}, {2}, {4}, {5}] + Q[{1, 3}, {4}, {2}, {5}] + Q[{1, 3}, {4}, {5}, {2}] + Q[{4}, {1, 3}, {2}, {5}] + Q[{4}, {1, 3}, {5}, {2}] + Q[{4}, {5}, {1, 3}, {2}] sage: Q[[1, 3], [2]].coproduct() Q[] # Q[{1, 3}, {2}] + Q[{1, 2}] # Q[{1}] + Q[{1, 3}, {2}] # Q[]
REFERENCES:
Section 6 of [BerZab05]
- class Element#
Bases:
IndexedFreeModuleElement
- algebraic_complement()#
Return the image of the element
self
of \(WQSym\) under the algebraic complement involution.See
WQSymBases.ElementMethods.algebraic_complement()
for a definition of the involution and for examples.See also
EXAMPLES:
sage: WQSym = algebras.WQSym(ZZ) sage: Q = WQSym.Q() sage: Q[[1,2],[5,6],[3,4]].algebraic_complement() Q[{3, 4}, {1, 2, 5, 6}] + Q[{3, 4}, {5, 6}, {1, 2}] - Q[{3, 4, 5, 6}, {1, 2}] sage: Q[[3], [1, 2], [4]].algebraic_complement() Q[{1, 2, 4}, {3}] + Q[{4}, {1, 2}, {3}] - Q[{4}, {1, 2, 3}]
- coalgebraic_complement()#
Return the image of the element
self
of \(WQSym\) under the coalgebraic complement involution.See
WQSymBases.ElementMethods.coalgebraic_complement()
for a definition of the involution and for examples.See also
EXAMPLES:
sage: WQSym = algebras.WQSym(ZZ) sage: Q = WQSym.Q() sage: Q[[1,2],[5,6],[3,4]].coalgebraic_complement() Q[{1, 2, 5, 6}, {3, 4}] + Q[{5, 6}, {1, 2}, {3, 4}] - Q[{5, 6}, {1, 2, 3, 4}] sage: Q[[3], [1, 2], [4]].coalgebraic_complement() Q[{2}, {1, 3, 4}] + Q[{2}, {3, 4}, {1}] - Q[{2, 3, 4}, {1}]
- star_involution()#
Return the image of the element
self
of \(WQSym\) under the star involution.See
WQSymBases.ElementMethods.star_involution()
for a definition of the involution and for examples.EXAMPLES:
sage: WQSym = algebras.WQSym(ZZ) sage: Q = WQSym.Q() sage: Q[[1,2],[5,6],[3,4]].star_involution() Q[{3, 4}, {1, 2}, {5, 6}] sage: Q[[3], [1, 2], [4]].star_involution() Q[{1}, {3, 4}, {2}]
- coproduct_on_basis(x)#
Return the coproduct of
self
on the basis element indexed by the ordered set partitionx
.EXAMPLES:
sage: Q = algebras.WQSym(QQ).Q() sage: Q.coproduct(Q.one()) # indirect doctest Q[] # Q[] sage: Q.coproduct( Q([[1]]) ) # indirect doctest Q[] # Q[{1}] + Q[{1}] # Q[] sage: Q.coproduct( Q([[1,2]]) ) Q[] # Q[{1, 2}] + Q[{1, 2}] # Q[] sage: Q.coproduct( Q([[1], [2]]) ) Q[] # Q[{1}, {2}] + Q[{1}] # Q[{1}] + Q[{1}, {2}] # Q[] sage: Q[[1,2],[3],[4]].coproduct() Q[] # Q[{1, 2}, {3}, {4}] + Q[{1, 2}] # Q[{1}, {2}] + Q[{1, 2}, {3}] # Q[{1}] + Q[{1, 2}, {3}, {4}] # Q[]
- product_on_basis(x, y)#
Return the (associative) \(*\) product of the basis elements of the Q basis
self
indexed by the ordered set partitions \(x\) and \(y\).This is the shifted shuffle product of \(x\) and \(y\).
EXAMPLES:
sage: A = algebras.WQSym(QQ).Q() sage: x = OrderedSetPartition([[1],[2,3]]) sage: y = OrderedSetPartition([[1,2]]) sage: z = OrderedSetPartition([[1,2],[3]]) sage: A.product_on_basis(x, y) Q[{1}, {2, 3}, {4, 5}] + Q[{1}, {4, 5}, {2, 3}] + Q[{4, 5}, {1}, {2, 3}] sage: A.product_on_basis(x, z) Q[{1}, {2, 3}, {4, 5}, {6}] + Q[{1}, {4, 5}, {2, 3}, {6}] + Q[{1}, {4, 5}, {6}, {2, 3}] + Q[{4, 5}, {1}, {2, 3}, {6}] + Q[{4, 5}, {1}, {6}, {2, 3}] + Q[{4, 5}, {6}, {1}, {2, 3}] sage: A.product_on_basis(y, y) Q[{1, 2}, {3, 4}] + Q[{3, 4}, {1, 2}]
- some_elements()#
Return some elements of the word quasi-symmetric functions in the Q basis.
EXAMPLES:
sage: Q = algebras.WQSym(QQ).Q() sage: Q.some_elements() [Q[], Q[{1}], Q[{1, 2}], Q[] + 1/2*Q[{1}]]
- class StronglyFiner(alg)#
Bases:
WQSymBasis_abstract
The Phi basis of \(WQSym\).
We define a partial order \(\leq\) on the set of all ordered set partitions as follows: \(A \leq B\) if and only if \(A\) is strongly finer than \(B\) (see
is_strongly_finer()
for a definition of this).The Phi basis \((\Phi_P)_P\) is a basis of \(WQSym\) indexed by ordered set partitions, and is defined by
\[\Phi_P = \sum \mathbf{M}_W,\]where the sum is over ordered set partitions \(W\) satisfying \(W \leq P\).
Novelli and Thibon introduced this basis in [NovThi06] Section 2.7.2, and called it the quasi-ribbon basis. It later reappeared in [MeNoTh11] Section 4.3.2.
EXAMPLES:
sage: WQSym = algebras.WQSym(QQ) sage: M = WQSym.M(); Phi = WQSym.Phi() sage: Phi Word Quasi-symmetric functions over Rational Field in the Phi basis sage: Phi(M[[2,3],[1,4]]) Phi[{2}, {3}, {1}, {4}] - Phi[{2}, {3}, {1, 4}] - Phi[{2, 3}, {1}, {4}] + Phi[{2, 3}, {1, 4}] sage: Phi(M[[1,2],[3,4]]) Phi[{1}, {2}, {3}, {4}] - Phi[{1}, {2}, {3, 4}] - Phi[{1, 2}, {3}, {4}] + Phi[{1, 2}, {3, 4}] sage: M(Phi[[1,2],[3,4]]) M[{1}, {2}, {3}, {4}] + M[{1}, {2}, {3, 4}] + M[{1, 2}, {3}, {4}] + M[{1, 2}, {3, 4}] sage: M(Phi[[2,3],[1],[4]]) M[{2}, {3}, {1}, {4}] + M[{2, 3}, {1}, {4}] sage: M(Phi[[3], [2, 5], [1, 4]]) M[{3}, {2}, {5}, {1}, {4}] + M[{3}, {2}, {5}, {1, 4}] + M[{3}, {2, 5}, {1}, {4}] + M[{3}, {2, 5}, {1, 4}] sage: M(Phi[[1, 4], [2, 3], [5], [6]]) M[{1}, {4}, {2}, {3}, {5}, {6}] + M[{1}, {4}, {2, 3}, {5}, {6}] + M[{1, 4}, {2}, {3}, {5}, {6}] + M[{1, 4}, {2, 3}, {5}, {6}] sage: Phi[[1],] * Phi[[1, 3], [2]] Phi[{1, 2, 4}, {3}] + Phi[{2}, {1, 4}, {3}] + Phi[{2, 4}, {1, 3}] + Phi[{2, 4}, {3}, {1}] sage: Phi[[3, 5], [1, 4], [2]].coproduct() Phi[] # Phi[{3, 5}, {1, 4}, {2}] + Phi[{1}] # Phi[{4}, {1, 3}, {2}] + Phi[{1, 2}] # Phi[{1, 3}, {2}] + Phi[{2, 3}, {1}] # Phi[{2}, {1}] + Phi[{2, 4}, {1, 3}] # Phi[{1}] + Phi[{3, 5}, {1, 4}, {2}] # Phi[]
REFERENCES:
Section 2.7.2 of [NovThi06]
- class Element#
Bases:
IndexedFreeModuleElement
- algebraic_complement()#
Return the image of the element
self
of \(WQSym\) under the algebraic complement involution.See
WQSymBases.ElementMethods.algebraic_complement()
for a definition of the involution and for examples.See also
EXAMPLES:
sage: WQSym = algebras.WQSym(ZZ) sage: Phi = WQSym.Phi() sage: Phi[[1],[2,4],[3]].algebraic_complement() -Phi[{3}, {2}, {4}, {1}] + Phi[{3}, {2, 4}, {1}] + Phi[{3}, {4}, {2}, {1}] sage: Phi[[1],[2,3],[4]].algebraic_complement() -Phi[{4}, {2}, {3}, {1}] + Phi[{4}, {2, 3}, {1}] + Phi[{4}, {3}, {2}, {1}]
- coalgebraic_complement()#
Return the image of the element
self
of \(WQSym\) under the coalgebraic complement involution.See
WQSymBases.ElementMethods.coalgebraic_complement()
for a definition of the involution and for examples.See also
EXAMPLES:
sage: WQSym = algebras.WQSym(ZZ) sage: Phi = WQSym.Phi() sage: Phi[[1],[2],[3,4]].coalgebraic_complement() -Phi[{4}, {3}, {1}, {2}] + Phi[{4}, {3}, {1, 2}] + Phi[{4}, {3}, {2}, {1}] sage: Phi[[2],[1,4],[3]].coalgebraic_complement() -Phi[{3}, {1}, {4}, {2}] + Phi[{3}, {1, 4}, {2}] + Phi[{3}, {4}, {1}, {2}]
- star_involution()#
Return the image of the element
self
of \(WQSym\) under the star involution.See
WQSymBases.ElementMethods.star_involution()
for a definition of the involution and for examples.EXAMPLES:
sage: WQSym = algebras.WQSym(ZZ) sage: Phi = WQSym.Phi() sage: Phi[[1,2],[5,6],[3,4]].star_involution() Phi[{3, 4}, {1, 2}, {5, 6}] sage: Phi[[3], [1, 2], [4]].star_involution() Phi[{1}, {3, 4}, {2}]
- coproduct_on_basis(x)#
Return the coproduct of
self
on the basis element indexed by the ordered set partitionx
.The coproduct of the basis element \(\Phi_x\) indexed by an ordered set partition \(x\) of \([n]\) can be computed by the following formula ([NovThi06]):
\[\Delta \Phi_x = \sum \Phi_y \otimes \Phi_z ,\]where the sum ranges over all pairs \((y, z)\) of ordered set partitions \(y\) and \(z\) such that:
\(y\) and \(z\) are ordered set partitions of two complementary subsets of \([n]\);
\(x\) is obtained either by concatenating \(y\) and \(z\), or by first concatenating \(y\) and \(z\) and then merging the two “middle blocks” (i.e., the last block of \(y\) and the first block of \(z\)); in the latter case, the maximum of the last block of \(y\) has to be smaller than the minimum of the first block of \(z\) (so that when merging these blocks, their entries don’t need to be sorted).
EXAMPLES:
sage: Phi = algebras.WQSym(QQ).Phi() sage: Phi.coproduct(Phi.one()) # indirect doctest Phi[] # Phi[] sage: Phi.coproduct( Phi([[1]]) ) # indirect doctest Phi[] # Phi[{1}] + Phi[{1}] # Phi[] sage: Phi.coproduct( Phi([[1,2]]) ) Phi[] # Phi[{1, 2}] + Phi[{1}] # Phi[{1}] + Phi[{1, 2}] # Phi[] sage: Phi.coproduct( Phi([[1], [2]]) ) Phi[] # Phi[{1}, {2}] + Phi[{1}] # Phi[{1}] + Phi[{1}, {2}] # Phi[] sage: Phi[[1,2],[3],[4]].coproduct() Phi[] # Phi[{1, 2}, {3}, {4}] + Phi[{1}] # Phi[{1}, {2}, {3}] + Phi[{1, 2}] # Phi[{1}, {2}] + Phi[{1, 2}, {3}] # Phi[{1}] + Phi[{1, 2}, {3}, {4}] # Phi[]
- product_on_basis(x, y)#
Return the (associative) \(*\) product of the basis elements of the Phi basis
self
indexed by the ordered set partitions \(x\) and \(y\).This is obtained by the following algorithm (going back to [NovThi06]):
Let \(x\) be an ordered set partition of \([m]\), and \(y\) an ordered set partition of \([n]\). Transform \(x\) into a list \(u\) of all the \(m\) elements of \([m]\) by writing out each block of \(x\) (in increasing order) and putting bars between each two consecutive blocks; this is called a barred permutation. Do the same for \(y\), but also shift each entry of the resulting barred permutation by \(m\). Let \(v\) be the barred permutation of \([m+n] \setminus [m]\) thus obtained. Now, shuffle the two barred permutations \(u\) and \(v\) (ignoring the bars) in all the \(\binom{n+m}{n}\) possible ways. For each shuffle obtained, place bars between some entries of the shuffle, according to the following rule:
If two consecutive entries of the shuffle both come from \(u\), then place a bar between them if the corresponding entries of \(u\) had a bar between them.
If the first of two consecutive entries of the shuffle comes from \(v\) and the second from \(u\), then place a bar between them.
This results in a barred permutation of \([m+n]\). Transform it into an ordered set partition of \([m+n]\), by treating the bars as dividers separating consecutive blocks.
The product \(\Phi_x \Phi_y\) is the sum of \(\Phi_p\) with \(p\) ranging over all ordered set partitions obtained this way.
EXAMPLES:
sage: A = algebras.WQSym(QQ).Phi() sage: x = OrderedSetPartition([[1],[2,3]]) sage: y = OrderedSetPartition([[1,2]]) sage: z = OrderedSetPartition([[1,2],[3]]) sage: A.product_on_basis(x, y) Phi[{1}, {2, 3, 4, 5}] + Phi[{1}, {2, 4}, {3, 5}] + Phi[{1}, {2, 4, 5}, {3}] + Phi[{1, 4}, {2, 3, 5}] + Phi[{1, 4}, {2, 5}, {3}] + Phi[{1, 4, 5}, {2, 3}] + Phi[{4}, {1}, {2, 3, 5}] + Phi[{4}, {1}, {2, 5}, {3}] + Phi[{4}, {1, 5}, {2, 3}] + Phi[{4, 5}, {1}, {2, 3}] sage: A.product_on_basis(x, z) Phi[{1}, {2, 3, 4, 5}, {6}] + Phi[{1}, {2, 4}, {3, 5}, {6}] + Phi[{1}, {2, 4, 5}, {3, 6}] + Phi[{1}, {2, 4, 5}, {6}, {3}] + Phi[{1, 4}, {2, 3, 5}, {6}] + Phi[{1, 4}, {2, 5}, {3, 6}] + Phi[{1, 4}, {2, 5}, {6}, {3}] + Phi[{1, 4, 5}, {2, 3, 6}] + Phi[{1, 4, 5}, {2, 6}, {3}] + Phi[{1, 4, 5}, {6}, {2, 3}] + Phi[{4}, {1}, {2, 3, 5}, {6}] + Phi[{4}, {1}, {2, 5}, {3, 6}] + Phi[{4}, {1}, {2, 5}, {6}, {3}] + Phi[{4}, {1, 5}, {2, 3, 6}] + Phi[{4}, {1, 5}, {2, 6}, {3}] + Phi[{4}, {1, 5}, {6}, {2, 3}] + Phi[{4, 5}, {1}, {2, 3, 6}] + Phi[{4, 5}, {1}, {2, 6}, {3}] + Phi[{4, 5}, {1, 6}, {2, 3}] + Phi[{4, 5}, {6}, {1}, {2, 3}] sage: A.product_on_basis(y, y) Phi[{1, 2, 3, 4}] + Phi[{1, 3}, {2, 4}] + Phi[{1, 3, 4}, {2}] + Phi[{3}, {1, 2, 4}] + Phi[{3}, {1, 4}, {2}] + Phi[{3, 4}, {1, 2}]
- some_elements()#
Return some elements of the word quasi-symmetric functions in the Phi basis.
EXAMPLES:
sage: Phi = algebras.WQSym(QQ).Phi() sage: Phi.some_elements() [Phi[], Phi[{1}], Phi[{1, 2}], Phi[] + 1/2*Phi[{1}]]
- X#
alias of
Characteristic
- a_realization()#
Return a particular realization of
self
(the \(M\)-basis).EXAMPLES:
sage: WQSym = algebras.WQSym(QQ) sage: WQSym.a_realization() Word Quasi-symmetric functions over Rational Field in the Monomial basis
- options = Current options for WordQuasiSymmetricFunctions element - display: normal - objects: compositions#