Free Quasi-symmetric functions

AUTHORS:

  • Frédéric Chapoton, Darij Grinberg (2017)
class sage.combinat.fqsym.FQSymBases(base)

Bases: sage.categories.realizations.Category_realization_of_parent

The category of graded bases of \(FQSym\) indexed by permutations.

class ElementMethods
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

psi_involution(), star_involution()

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

omega_involution(), star_involution()

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

omega_involution(), psi_involution()

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 of self); 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
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 \(\prec\).

See also

product(), succ()

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 \(\succ\).

See also

product(), prec()

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]
succ_by_coercion(x, y)

Return \(x \succ y\), computed using coercion to the F-basis.

See succ() for the definition of the objects involved.

EXAMPLES:

sage: G = algebras.FQSym(ZZ).G()
sage: G.succ(G([1]), G([2, 3, 1])) # indirect doctest
G[2, 3, 4, 1] + G[3, 2, 4, 1] + G[4, 2, 3, 1]
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: sage.combinat.free_module.CombinatorialFreeModule, sage.misc.bindable_class.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: sage.structure.unique_representation.UniqueRepresentation, sage.structure.parent.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: sage.combinat.fqsym.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: sage.modules.with_basis.indexed_element.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 of self); 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

product_on_basis(), succ_product_on_basis()

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 \(y\).

See also

succ_product_on_basis(), prec_product_on_basis()

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: sage.combinat.fqsym.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: sage.combinat.fqsym.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: sage.modules.with_basis.indexed_element.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