Free Quasi-symmetric functions#
AUTHORS:
Frédéric Chapoton, Darij Grinberg (2017)
- class sage.combinat.fqsym.FQSymBases(base)#
Bases:
Category_realization_of_parent
The category of graded bases of \(FQSym\) indexed by permutations.
- class ElementMethods#
Bases:
object
- omega_involution()#
Return the image of the element
self
of \(FQSym\) under the omega involution.The \(\omega\) involution is defined as the linear map \(FQSym \to FQSym\) that sends each basis element \(F_u\) of the F-basis of \(FQSym\) to the basis element \(F_{u \circ w_0}\), where \(w_0\) is the longest word (i.e., \(w_0(i) = n + 1 - i\)) in the symmetric group \(S_n\) that contains \(u\). The \(\omega\) involution is a graded algebra automorphism and a coalgebra anti-automorphism of \(FQSym\). Every permutation \(u \in S_n\) satisfies
\[\omega(F_u) = F_{u \circ w_0}, \qquad \omega(G_u) = G_{w_0 \circ u},\]where standard notations for classical bases of \(FQSym\) are being used (that is, \(F\) for the F-basis, and \(G\) for the G-basis). In other words, writing permutations in one-line notation, we have
\[\omega(F_{(u_1, u_2, \ldots, u_n)}) = F_{(u_n, u_{n-1}, \ldots, u_1)}, \qquad \omega(G_{(u_1, u_2, \ldots, u_n)}) = G_{(n+1-u_1, n+1-u_2, \ldots, n+1-u_n)}.\]If we also consider the \(\omega\) involution (
omega_involution()
) of the quasisymmetric functions (by slight abuse of notation), and if we let \(\pi\) be the canonical projection \(FQSym \to QSym\), then \(\pi \circ \omega = \omega \circ \pi\).Additionally, consider the \(\psi\) involution (
psi_involution()
) of the noncommutative symmetric functions, and if we let \(\iota\) be the canonical inclusion \(NSym \to FQSym\), then \(\omega \circ \iota = \iota \circ \psi\).Todo
Duality?
See also
EXAMPLES:
sage: FQSym = algebras.FQSym(ZZ) sage: F = FQSym.F() sage: F[[2,3,1]].omega_involution() F[1, 3, 2] sage: (3*F[[1]] - 4*F[[]] + 5*F[[1,2]]).omega_involution() -4*F[] + 3*F[1] + 5*F[2, 1] sage: G = FQSym.G() sage: G[[2,3,1]].omega_involution() G[2, 1, 3] sage: M = FQSym.M() sage: M[[2,3,1]].omega_involution() -M[1, 2, 3] - M[2, 1, 3] - M[3, 1, 2]
The omega involution is an algebra homomorphism:
sage: (F[1,2] * F[1]).omega_involution() F[2, 1, 3] + F[2, 3, 1] + F[3, 2, 1] sage: F[1,2].omega_involution() * F[1].omega_involution() F[2, 1, 3] + F[2, 3, 1] + F[3, 2, 1]
The omega involution intertwines the antipode and the inverse of the antipode:
sage: all( F(I).antipode().omega_involution().antipode() ....: == F(I).omega_involution() ....: for I in Permutations(4) ) True
Testing the \(\pi \circ \omega = \omega \circ \pi\) relation noticed above:
sage: all( M[I].omega_involution().to_qsym() ....: == M[I].to_qsym().omega_involution() ....: for I in Permutations(4) ) True
Testing the \(\omega \circ \iota = \iota \circ \psi\) relation:
sage: NSym = NonCommutativeSymmetricFunctions(ZZ) sage: S = NSym.S() sage: all( S[I].psi_involution().to_fqsym() == S[I].to_fqsym().omega_involution() ....: for I in Compositions(4) ) True
Todo
Check further commutative squares.
- psi_involution()#
Return the image of the element
self
of \(FQSym\) under the psi involution.The \(\psi\) involution is defined as the linear map \(FQSym \to FQSym\) that sends each basis element \(F_u\) of the F-basis of \(FQSym\) to the basis element \(F_{w_0 \circ u}\), where \(w_0\) is the longest word (i.e., \(w_0(i) = n + 1 - i\)) in the symmetric group \(S_n\) that contains \(u\). The \(\psi\) involution is a graded coalgebra automorphism and an algebra anti-automorphism of \(FQSym\). Every permutation \(u \in S_n\) satisfies
\[\psi(F_u) = F_{w_0 \circ u}, \qquad \psi(G_u) = G_{u \circ w_0},\]where standard notations for classical bases of \(FQSym\) are being used (that is, \(F\) for the F-basis, and \(G\) for the G-basis). In other words, writing permutations in one-line notation, we have
\[\psi(F_{(u_1, u_2, \ldots, u_n)}) = F_{(n+1-u_1, n+1-u_2, \ldots, n+1-u_n)}, \qquad \psi(G_{(u_1, u_2, \ldots, u_n)}) = G_{(u_n, u_{n-1}, \ldots, u_1)}.\]If we also consider the \(\psi\) involution (
psi_involution()
) of the quasisymmetric functions (by slight abuse of notation), and if we let \(\pi\) be the canonical projection \(FQSym \to QSym\), then \(\pi \circ \psi = \psi \circ \pi\).Additionally, consider the \(\omega\) involution (
omega_involution()
) of the noncommutative symmetric functions, and if we let \(\iota\) be the canonical inclusion \(NSym \to FQSym\), then \(\psi \circ \iota = \iota \circ \omega\).Todo
Duality?
See also
EXAMPLES:
sage: FQSym = algebras.FQSym(ZZ) sage: F = FQSym.F() sage: F[[2,3,1]].psi_involution() F[2, 1, 3] sage: (3*F[[1]] - 4*F[[]] + 5*F[[1,2]]).psi_involution() -4*F[] + 3*F[1] + 5*F[2, 1] sage: G = FQSym.G() sage: G[[2,3,1]].psi_involution() G[1, 3, 2] sage: M = FQSym.M() sage: M[[2,3,1]].psi_involution() -M[1, 2, 3] - M[1, 3, 2] - M[2, 3, 1]
The \(\psi\) involution intertwines the antipode and the inverse of the antipode:
sage: all( F(I).antipode().psi_involution().antipode() ....: == F(I).psi_involution() ....: for I in Permutations(4) ) True
Testing the \(\pi \circ \psi = \psi \circ \pi\) relation above:
sage: all( M[I].psi_involution().to_qsym() ....: == M[I].to_qsym().psi_involution() ....: for I in Permutations(4) ) True
Testing the \(\psi \circ \iota = \iota \circ \omega\) relation:
sage: NSym = NonCommutativeSymmetricFunctions(ZZ) sage: S = NSym.S() sage: all( S[I].omega_involution().to_fqsym() == S[I].to_fqsym().psi_involution() ....: for I in Compositions(4) ) True
Todo
Check further commutative squares.
- star_involution()#
Return the image of the element
self
of \(FQSym\) under the star involution.The star involution is defined as the linear map \(FQSym \to FQSym\) that sends each basis element \(F_u\) of the F-basis of \(FQSym\) to the basis element \(F_{w_0 \circ u \circ w_0}\), where \(w_0\) is the longest word (i.e., \(w_0(i) = n + 1 - i\)) in the symmetric group \(S_n\) that contains \(u\). The star involution is a graded Hopf algebra anti-automorphism of \(FQSym\). It is denoted by \(f \mapsto f^*\). Every permutation \(u \in S_n\) satisfies
\[(F_u)^* = F_{w_0 \circ u \circ w_0}, \qquad (G_u)^* = G_{w_0 \circ u \circ w_0}, \qquad (\mathcal{M}_u)^* = \mathcal{M}_{w_0 \circ u \circ w_0},\]where standard notations for classical bases of \(FQSym\) are being used (that is, \(F\) for the F-basis, \(G\) for the G-basis, and \(\mathcal{M}\) for the Monomial basis). In other words, writing permutations in one-line notation, we have
\[(F_{(u_1, u_2, \ldots, u_n)})^* = F_{(n+1-u_n, n+1-u_{n-1}, \ldots, n+1-u_1)}, \qquad (G_{(u_1, u_2, \ldots, u_n)})^* = G_{(n+1-u_n, n+1-u_{n-1}, \ldots, n+1-u_1)},\]and
\[(\mathcal{M}_{(u_1, u_2, \ldots, u_n)})^* = \mathcal{M}_{(n+1-u_n, n+1-u_{n-1}, \ldots, n+1-u_1)}.\]Let us denote the star involution by \((\ast)\) as well.
If we also denote by \((\ast)\) the star involution of of the quasisymmetric functions (
star_involution()
) and if we let \(\pi : FQSym \to QSym\) be the canonical projection then \(\pi \circ (\ast) = (\ast) \circ \pi\). Similar for the noncommutative symmetric functions (star_involution()
) with \(\pi : NSym \to FQSym\) being the canonical inclusion and the word quasisymmetric functions (star_involution()
) with \(\pi : FQSym \to WQSym\) the canonical inclusion.Todo
Duality?
See also
EXAMPLES:
sage: FQSym = algebras.FQSym(ZZ) sage: F = FQSym.F() sage: F[[2,3,1]].star_involution() F[3, 1, 2] sage: (3*F[[1]] - 4*F[[]] + 5*F[[1,2]]).star_involution() -4*F[] + 3*F[1] + 5*F[1, 2] sage: G = FQSym.G() sage: G[[2,3,1]].star_involution() G[3, 1, 2] sage: M = FQSym.M() sage: M[[2,3,1]].star_involution() M[3, 1, 2]
The star involution commutes with the antipode:
sage: all( F(I).antipode().star_involution() ....: == F(I).star_involution().antipode() ....: for I in Permutations(4) ) True
Testing the \(\pi \circ (\ast) = (\ast) \circ \pi\) relation:
sage: all( M[I].star_involution().to_qsym() ....: == M[I].to_qsym().star_involution() ....: for I in Permutations(4) ) True
Similar for \(NSym\):
sage: NSym = NonCommutativeSymmetricFunctions(ZZ) sage: S = NSym.S() sage: all( S[I].star_involution().to_fqsym() == S[I].to_fqsym().star_involution() ....: for I in Compositions(4) ) True
Similar for \(WQSym\):
sage: WQSym = algebras.WQSym(ZZ) sage: all( F(I).to_wqsym().star_involution() ....: == F(I).star_involution().to_wqsym() ....: for I in Permutations(4) ) True
Todo
Check further commutative squares.
- to_qsym()#
Return the image of
self
under the canonical projection \(FQSym \to QSym\).The canonical projection \(FQSym \to QSym\) is a surjective homomorphism of Hopf algebras. It sends a basis element \(F_w\) of \(FQSym\) to the basis element \(F_{\operatorname{Comp} w}\) of the fundamental basis of \(QSym\), where \(\operatorname{Comp} w\) stands for the descent composition (
sage.combinat.permutation.Permutation.descents_composition()
) of the permutation \(w\).See also
QuasiSymmetricFunctions
for a definition of \(QSym\).EXAMPLES:
sage: G = algebras.FQSym(QQ).G() sage: x = G[1, 3, 2] sage: x.to_qsym() F[2, 1] sage: G[2, 3, 1].to_qsym() F[1, 2] sage: F = algebras.FQSym(QQ).F() sage: F[2, 3, 1].to_qsym() F[2, 1] sage: (F[2, 3, 1] + F[1, 3, 2] + F[1, 2, 3]).to_qsym() 2*F[2, 1] + F[3] sage: F2 = algebras.FQSym(GF(2)).F() sage: F2[2, 3, 1].to_qsym() F[2, 1] sage: (F2[2, 3, 1] + F2[1, 3, 2] + F2[1, 2, 3]).to_qsym() F[3]
- to_symmetric_group_algebra(n=None)#
Return the element of a symmetric group algebra corresponding to the element
self
of \(FQSym\).INPUT:
n
– integer (default: the maximal degree ofself
); the rank of the target symmetric group algebra
EXAMPLES:
sage: A = algebras.FQSym(QQ).G() sage: x = A([1,3,2,4]) + 5/2 * A([2,3,4,1]) sage: x.to_symmetric_group_algebra() [1, 3, 2, 4] + 5/2*[4, 1, 2, 3]
- to_wqsym()#
Return the image of
self
under the canonical inclusion map \(FQSym \to WQSym\).The canonical inclusion map \(FQSym \to WQSym\) is an injective homomorphism of Hopf algebras. It sends a basis element \(G_w\) of \(FQSym\) to the sum of basis elements \(\mathbf{M}_u\) of \(WQSym\), where \(u\) ranges over all packed words whose standardization is \(w\).
See also
WordQuasiSymmetricFunctions
for a definition of \(WQSym\).EXAMPLES:
sage: G = algebras.FQSym(QQ).G() sage: x = G[1, 3, 2] sage: x.to_wqsym() M[{1}, {3}, {2}] + M[{1, 3}, {2}] sage: G[1, 2].to_wqsym() M[{1}, {2}] + M[{1, 2}] sage: F = algebras.FQSym(QQ).F() sage: F[3, 1, 2].to_wqsym() M[{3}, {1}, {2}] + M[{3}, {1, 2}] sage: G[2, 3, 1].to_wqsym() M[{3}, {1}, {2}] + M[{3}, {1, 2}]
- class ParentMethods#
Bases:
object
- basis(degree=None)#
The basis elements (optionally: of the specified degree).
OUTPUT: Family
EXAMPLES:
sage: FQSym = algebras.FQSym(QQ) sage: G = FQSym.G() sage: G.basis() Lazy family (Term map from Standard permutations to Free Quasi-symmetric functions over Rational Field in the G basis(i))_{i in Standard permutations} sage: G.basis().keys() Standard permutations sage: G.basis(degree=3).keys() Standard permutations of 3 sage: G.basis(degree=3).list() [G[1, 2, 3], G[1, 3, 2], G[2, 1, 3], G[2, 3, 1], G[3, 1, 2], G[3, 2, 1]]
- from_symmetric_group_algebra(x)#
Return the element of \(FQSym\) corresponding to the element \(x\) of a symmetric group algebra.
EXAMPLES:
sage: A = algebras.FQSym(QQ).F() sage: SGA4 = SymmetricGroupAlgebra(QQ, 4) sage: x = SGA4([1,3,2,4]) + 5/2 * SGA4([1,2,4,3]) sage: A.from_symmetric_group_algebra(x) 5/2*F[1, 2, 4, 3] + F[1, 3, 2, 4] sage: A.from_symmetric_group_algebra(SGA4.zero()) 0
- is_commutative()#
Return whether this \(FQSym\) is commutative.
EXAMPLES:
sage: F = algebras.FQSym(ZZ).F() sage: F.is_commutative() False
- is_field(proof=True)#
Return whether this \(FQSym\) is a field.
EXAMPLES:
sage: F = algebras.FQSym(QQ).F() sage: F.is_field() False
- one_basis()#
Return the index of the unit.
EXAMPLES:
sage: A = algebras.FQSym(QQ).F() sage: A.one_basis() []
- prec()#
Return the \(\prec\) product.
On the F-basis of
FQSym
, this product is determined by \(F_x \prec F_y = \sum F_z\), where the sum ranges over all \(z\) in the shifted shuffle of \(x\) and \(y\) with the additional condition that the first letter of the result comes from \(x\).The usual symbol for this operation is \(\prec\).
EXAMPLES:
sage: A = algebras.FQSym(QQ).F() sage: x = A([2,1]) sage: A.prec(x, x) F[2, 1, 4, 3] + F[2, 4, 1, 3] + F[2, 4, 3, 1] sage: y = A([2,1,3]) sage: A.prec(x, y) F[2, 1, 4, 3, 5] + F[2, 4, 1, 3, 5] + F[2, 4, 3, 1, 5] + F[2, 4, 3, 5, 1] sage: A.prec(y, x) F[2, 1, 3, 5, 4] + F[2, 1, 5, 3, 4] + F[2, 1, 5, 4, 3] + F[2, 5, 1, 3, 4] + F[2, 5, 1, 4, 3] + F[2, 5, 4, 1, 3]
- prec_by_coercion(x, y)#
Return \(x \prec y\), computed using coercion to the F-basis.
See
prec()
for the definition of the objects involved.EXAMPLES:
sage: G = algebras.FQSym(ZZ).G() sage: a = G([1]) sage: b = G([2, 3, 1]) sage: G.prec(a, b) + G.succ(a, b) == a * b # indirect doctest True
- some_elements()#
Return some elements of the free quasi-symmetric functions.
EXAMPLES:
sage: A = algebras.FQSym(QQ) sage: F = A.F() sage: F.some_elements() [F[], F[1], F[1, 2] + F[2, 1], F[] + F[1, 2] + F[2, 1]] sage: G = A.G() sage: G.some_elements() [G[], G[1], G[1, 2] + G[2, 1], G[] + G[1, 2] + G[2, 1]] sage: M = A.M() sage: M.some_elements() [M[], M[1], M[1, 2] + 2*M[2, 1], M[] + M[1, 2] + 2*M[2, 1]]
- succ()#
Return the \(\succ\) product.
On the F-basis of
FQSym
, this product is determined by \(F_x \succ F_y = \sum F_z\), where the sum ranges over all \(z\) in the shifted shuffle of \(x\) and \(y\) with the additional condition that the first letter of the result comes from \(y\).The usual symbol for this operation is \(\succ\).
EXAMPLES:
sage: A = algebras.FQSym(QQ).F() sage: x = A([1]) sage: A.succ(x, x) F[2, 1] sage: y = A([3,1,2]) sage: A.succ(x, y) F[4, 1, 2, 3] + F[4, 2, 1, 3] + F[4, 2, 3, 1] sage: A.succ(y, x) F[4, 3, 1, 2]
- super_categories()#
The super categories of
self
.EXAMPLES:
sage: from sage.combinat.fqsym import FQSymBases sage: FQSym = algebras.FQSym(ZZ) sage: bases = FQSymBases(FQSym) sage: bases.super_categories() [Category of realizations of Free 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]
- class sage.combinat.fqsym.FQSymBasis_abstract(alg)#
Bases:
CombinatorialFreeModule
,BindableClass
Abstract base class for bases of FQSym.
This must define two attributes:
_prefix
– the basis prefix_basis_name
– the name of the basis and must match one of the names that the basis can be constructed from FQSym
- an_element()#
Return an element of
self
.EXAMPLES:
sage: A = algebras.FQSym(QQ) sage: F = A.F() sage: F.an_element() F[1] + 2*F[1, 2] + 2*F[2, 1] sage: G = A.G() sage: G.an_element() G[1] + 2*G[1, 2] + 2*G[2, 1] sage: M = A.M() sage: M.an_element() M[1] + 2*M[1, 2] + 4*M[2, 1]
- class sage.combinat.fqsym.FreeQuasisymmetricFunctions(R)#
Bases:
UniqueRepresentation
,Parent
The free quasi-symmetric functions.
The Hopf algebra \(FQSym\) of free quasi-symmetric functions over a commutative ring \(R\) is the free \(R\)-module with basis indexed by all permutations (i.e., the indexing set is the disjoint union of all symmetric groups). Its product is determined by the shifted shuffles of two permutations, whereas its coproduct is given by splitting a permutation (regarded as a word) into two (at every possible point) and standardizing the two pieces. This Hopf algebra was introduced in [MR]. See [GriRei18] (Chapter 8) for a treatment using modern notations.
In more detail: For each \(n \geq 0\), consider the symmetric group \(S_n\). Let \(S\) be the disjoint union of the \(S_n\) over all \(n \geq 0\). Then, \(FQSym\) is the free \(R\)-module with basis \((F_w)_{w \in S}\). This \(R\)-module is graded, with the \(n\)-th graded component being spanned by all \(F_w\) for \(w \in S_n\). A multiplication is defined on \(FQSym\) as follows: For any two permutations \(u \in S_k\) and \(v \in S_l\), we set
\[F_u F_v = \sum F_w ,\]where the sum is over all shuffles of \(u\) with \(v[k]\). Here, the permutations \(u\) and \(v\) are regarded as words (by writing them in one-line notation), and \(v[k]\) means the word obtained from \(v\) by increasing each letter by \(k\) (for example, \((1,4,2,3)[5] = (6,9,7,8)\)); and the shuffles \(w\) are translated back into permutations. This defines an associative multiplication on \(FQSym\); its unity is \(F_e\), where \(e\) is the identity permutation in \(S_0\).
In Section 1.3 of [AguSot05], Aguiar and Sottile construct a different basis of \(FQSym\). Their basis, called the monomial basis and denoted by \((\mathcal{M}_u)\), is also indexed by permutations. It is connected to the above F-basis by the relation
\[F_u = \sum_v \mathcal{M}_v ,\]where the sum ranges over all permutations \(v\) such that each inversion of \(u\) is an inversion of \(v\). (An inversion of a permutation \(w\) means a pair \((i, j)\) of positions satisfying \(i < j\) and \(w(i) > w(j)\).) The above relation yields a unitriangular change-of-basis matrix, and thus can be used to compute the \(\mathcal{M}_u\) by Mobius inversion.
Another classical basis of \(FQSym\) is \((G_w)_{w \in S}\), where \(G_w = F_{w^{-1}}\). This is just a relabeling of the basis \((F_w)_{w \in S}\), but is a more natural choice from some viewpoints.
The algebra \(FQSym\) is often identified with (“realized 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. Consider the \(R\)-linear map that sends each \(G_u\) to the sum of all words whose standardization (also known as “standard permutation”; see
standard_permutation()
) is \(u\). This map is an injective \(R\)-algebra homomorphism, and thus embeds \(FQSym\) into the latter ring.As an associative algebra, \(FQSym\) has the richer structure of a dendriform algebra. This means that the associative product
*
is decomposed as a sum of two binary operations\[x y = x \succ y + x \prec y\]that satisfy the axioms:
\[(x \succ y) \prec z = x \succ (y \prec z),\]\[(x \prec y) \prec z = x \prec (y z),\]\[(x y) \succ z = x \succ (y \succ z).\]These two binary operations are defined similarly to the (associative) product above: We set
\[F_u \prec F_v = \sum F_w ,\]where the sum is now over all shuffles of \(u\) with \(v[k]\) whose first letter is taken from \(u\) (rather than from \(v[k]\)). Similarly,
\[F_u \succ F_v = \sum F_w ,\]where the sum is over all remaining shuffles of \(u\) with \(v[k]\).
Todo
Decide what \(1 \prec 1\) and \(1 \succ 1\) are.
Note
The usual binary operator
*
is used for the associative product.EXAMPLES:
sage: F = algebras.FQSym(ZZ).F() sage: x,y,z = F([1]), F([1,2]), F([1,3,2]) sage: (x * y) * z F[1, 2, 3, 4, 6, 5] + ...
The product of \(FQSym\) is associative:
sage: x * (y * z) == (x * y) * z True
The associative product decomposes into two parts:
sage: x * y == F.prec(x, y) + F.succ(x, y) True
The axioms of a dendriform algebra hold:
sage: F.prec(F.succ(x, y), z) == F.succ(x, F.prec(y, z)) True sage: F.prec(F.prec(x, y), z) == F.prec(x, y * z) True sage: F.succ(x * y, z) == F.succ(x, F.succ(y, z)) True
\(FQSym\) is also known as the Malvenuto-Reutenauer algebra:
sage: algebras.MalvenutoReutenauer(ZZ) Free Quasi-symmetric functions over Integer Ring
REFERENCES:
- class F(alg)#
Bases:
FQSymBasis_abstract
The F-basis of \(FQSym\).
This is the basis \((F_w)\), with \(w\) ranging over all permutations. See the documentation of
FreeQuasisymmetricFunctions
for details.EXAMPLES:
sage: FQSym = algebras.FQSym(QQ) sage: FQSym.F() Free Quasi-symmetric functions over Rational Field in the F basis
- class Element#
Bases:
IndexedFreeModuleElement
- to_symmetric_group_algebra(n=None)#
Return the element of a symmetric group algebra corresponding to the element
self
of \(FQSym\).INPUT:
n
– integer (default: the maximal degree ofself
); the rank of the target symmetric group algebra
EXAMPLES:
sage: A = algebras.FQSym(QQ).F() sage: x = A([1,3,2,4]) + 5/2 * A([1,2,4,3]) sage: x.to_symmetric_group_algebra() 5/2*[1, 2, 4, 3] + [1, 3, 2, 4] sage: x.to_symmetric_group_algebra(n=7) 5/2*[1, 2, 4, 3, 5, 6, 7] + [1, 3, 2, 4, 5, 6, 7] sage: a = A.zero().to_symmetric_group_algebra(); a 0 sage: parent(a) Symmetric group algebra of order 0 over Rational Field sage: y = A([1,3,2,4]) + 5/2 * A([2,1]) sage: y.to_symmetric_group_algebra() [1, 3, 2, 4] + 5/2*[2, 1, 3, 4] sage: y.to_symmetric_group_algebra(6) [1, 3, 2, 4, 5, 6] + 5/2*[2, 1, 3, 4, 5, 6]
- coproduct_on_basis(x)#
Return the coproduct of \(F_{\sigma}\) for \(\sigma\) a permutation (here, \(\sigma\) is
x
).EXAMPLES:
sage: A = algebras.FQSym(QQ).F() sage: x = A([1]) sage: ascii_art(A.coproduct(A.one())) # indirect doctest 1 # 1 sage: ascii_art(A.coproduct(x)) # indirect doctest 1 # F + F # 1 [1] [1] sage: A = algebras.FQSym(QQ).F() sage: x, y, z = A([1]), A([2,1]), A([3,2,1]) sage: A.coproduct(z) F[] # F[3, 2, 1] + F[1] # F[2, 1] + F[2, 1] # F[1] + F[3, 2, 1] # F[]
- degree_on_basis(t)#
Return the degree of a permutation in the algebra of free quasi-symmetric functions.
This is the size of the permutation (i.e., the \(n\) for which the permutation belongs to \(S_n\)).
EXAMPLES:
sage: A = algebras.FQSym(QQ).F() sage: u = Permutation([2,1]) sage: A.degree_on_basis(u) 2
- prec_product_on_basis(x, y)#
Return the \(\prec\) product of two permutations.
This is the shifted shuffle of \(x\) and \(y\) with the additional condition that the first letter of the result comes from \(x\).
The usual symbol for this operation is \(\prec\).
See also
EXAMPLES:
sage: A = algebras.FQSym(QQ).F() sage: x = Permutation([1,2]) sage: A.prec_product_on_basis(x, x) F[1, 2, 3, 4] + F[1, 3, 2, 4] + F[1, 3, 4, 2] sage: y = Permutation([]) sage: A.prec_product_on_basis(x, y) == A(x) True sage: A.prec_product_on_basis(y, x) == 0 True
- product_on_basis(x, y)#
Return the \(*\) associative product of two permutations.
This is the shifted shuffle of \(x\) and \(y\).
EXAMPLES:
sage: A = algebras.FQSym(QQ).F() sage: x = Permutation([1]) sage: A.product_on_basis(x, x) F[1, 2] + F[2, 1]
- succ_product_on_basis(x, y)#
Return the \(\succ\) product of two permutations.
This is the shifted shuffle of \(x\) and \(y\) with the additional condition that the first letter of the result comes from \(y\).
The usual symbol for this operation is \(\succ\).
See also
EXAMPLES:
sage: A = algebras.FQSym(QQ).F() sage: x = Permutation([1,2]) sage: A.succ_product_on_basis(x, x) F[3, 1, 2, 4] + F[3, 1, 4, 2] + F[3, 4, 1, 2] sage: y = Permutation([]) sage: A.succ_product_on_basis(x, y) == 0 True sage: A.succ_product_on_basis(y, x) == A(x) True
- class G(alg)#
Bases:
FQSymBasis_abstract
The G-basis of \(FQSym\).
This is the basis \((G_w)\), with \(w\) ranging over all permutations. See the documentation of
FreeQuasisymmetricFunctions
for details.EXAMPLES:
sage: FQSym = algebras.FQSym(QQ) sage: G = FQSym.G(); G Free Quasi-symmetric functions over Rational Field in the G basis sage: G([3, 1, 2]).coproduct() G[] # G[3, 1, 2] + G[1] # G[2, 1] + G[1, 2] # G[1] + G[3, 1, 2] # G[] sage: G([3, 1, 2]) * G([2, 1]) G[3, 1, 2, 5, 4] + G[4, 1, 2, 5, 3] + G[4, 1, 3, 5, 2] + G[4, 2, 3, 5, 1] + G[5, 1, 2, 4, 3] + G[5, 1, 3, 4, 2] + G[5, 1, 4, 3, 2] + G[5, 2, 3, 4, 1] + G[5, 2, 4, 3, 1] + G[5, 3, 4, 2, 1]
- degree_on_basis(t)#
Return the degree of a permutation in the algebra of free quasi-symmetric functions.
This is the size of the permutation (i.e., the \(n\) for which the permutation belongs to \(S_n\)).
EXAMPLES:
sage: A = algebras.FQSym(QQ).G() sage: u = Permutation([2,1]) sage: A.degree_on_basis(u) 2
- class M(alg)#
Bases:
FQSymBasis_abstract
The M-basis of \(FQSym\).
This is the Monomial basis \((\mathcal{M}_w)\), with \(w\) ranging over all permutations. See the documentation of
FQSym
for details.EXAMPLES:
sage: FQSym = algebras.FQSym(QQ) sage: M = FQSym.M(); M Free Quasi-symmetric functions over Rational Field in the Monomial basis sage: M([3, 1, 2]).coproduct() M[] # M[3, 1, 2] + M[1] # M[1, 2] + M[3, 1, 2] # M[] sage: M([3, 2, 1]).coproduct() M[] # M[3, 2, 1] + M[1] # M[2, 1] + M[2, 1] # M[1] + M[3, 2, 1] # M[] sage: M([1, 2]) * M([1]) M[1, 2, 3] + 2*M[1, 3, 2] + M[2, 3, 1] + M[3, 1, 2]
- class Element#
Bases:
IndexedFreeModuleElement
- star_involution()#
Return the image of the element
self
of \(FQSym\) under the star involution.See
FQSymBases.ElementMethods.star_involution()
for a definition of the involution and for examples.See also
omega_involution()
,psi_involution()
EXAMPLES:
sage: FQSym = algebras.FQSym(ZZ) sage: M = FQSym.M() sage: M[[2,3,1]].star_involution() M[3, 1, 2] sage: M[[]].star_involution() M[]
- coproduct_on_basis(x)#
Return the coproduct of \(\mathcal{M}_{\sigma}\) for \(\sigma\) a permutation (here, \(\sigma\) is
x
).This uses Theorem 3.1 in [AguSot05].
EXAMPLES:
sage: M = algebras.FQSym(QQ).M() sage: x = M([1]) sage: ascii_art(M.coproduct(M.one())) # indirect doctest 1 # 1 sage: ascii_art(M.coproduct(x)) # indirect doctest 1 # M + M # 1 [1] [1] sage: M.coproduct(M([2, 1, 3])) M[] # M[2, 1, 3] + M[2, 1, 3] # M[] sage: M.coproduct(M([2, 3, 1])) M[] # M[2, 3, 1] + M[1, 2] # M[1] + M[2, 3, 1] # M[] sage: M.coproduct(M([3, 2, 1])) M[] # M[3, 2, 1] + M[1] # M[2, 1] + M[2, 1] # M[1] + M[3, 2, 1] # M[] sage: M.coproduct(M([3, 4, 2, 1])) M[] # M[3, 4, 2, 1] + M[1, 2] # M[2, 1] + M[2, 3, 1] # M[1] + M[3, 4, 2, 1] # M[] sage: M.coproduct(M([3, 4, 1, 2])) M[] # M[3, 4, 1, 2] + M[1, 2] # M[1, 2] + M[3, 4, 1, 2] # M[]
- degree_on_basis(t)#
Return the degree of a permutation in the algebra of free quasi-symmetric functions.
This is the size of the permutation (i.e., the \(n\) for which the permutation belongs to \(S_n\)).
EXAMPLES:
sage: A = algebras.FQSym(QQ).M() sage: u = Permutation([2,1]) sage: A.degree_on_basis(u) 2
- a_realization()#
Return a particular realization of
self
(the F-basis).EXAMPLES:
sage: FQSym = algebras.FQSym(QQ) sage: FQSym.a_realization() Free Quasi-symmetric functions over Rational Field in the F basis