REFERENCES:
[Ges] | I. Gessel, Multipartite P-partitions and inner products of skew Schur functions, Contemp. Math. 34 (1984), 289-301. http://people.brandeis.edu/~gessel/homepage/papers/multipartite.pdf |
[MR] | C. Malvenuto and C. Reutenauer, Duality between quasi-symmetric functions and the Solomon descent algebra, J. Algebra 177 (1995), no. 3, 967-982. http://www.mat.uniroma1.it/people/malvenuto/Duality.pdf |
[Reiner2013] | Victor Reiner, Hopf algebras in combinatorics, 17 January 2013. http://www.math.umn.edu/~reiner/Classes/HopfComb.pdf |
[Mal1993] | Claudia Malvenuto, Produits et coproduits des fonctions quasi-symetriques et de l’algebre des descentes, thesis, November 1993. http://www1.mat.uniroma1.it/people/malvenuto/Thesis.pdf |
[Haz2004] | (1, 2, 3, 4) Michiel Hazewinkel, Explicit polynomial generators for the ring of quasisymmetric functions over the integers. Arxiv math/0410366v1 |
[Rad1979] | David E. Radford, A natural ring basis for the shuffle algebra and an application to group schemes, J. Algebra 58 (1979), 432-454. |
AUTHOR:
Bases: sage.structure.unique_representation.UniqueRepresentation, sage.structure.parent.Parent
The Hopf algebra of quasisymmetric functions.
Let be a commutative ring with unity.
The
-algebra of quasi-symmetric functions may be realized as an
-subalgebra of the ring of power series in countably many
variables
. It consists of those
formal power series
which are degree-bounded (i. e., the degrees
of all monomials occuring with nonzero coefficient in
are bounded
from above, although the bound can depend on
) and satisfy the
following condition: For every tuple
of
positive integers, the coefficient of the monomial
in
is the same
for all strictly increasing sequences
of
positive integers. (In other words, the coefficient of a monomial in
depends only on the sequence of nonzero exponents in the monomial. If
“sequence” were to be replaced by “multiset” here, we would obtain
the definition of a symmetric function.)
The -algebra of quasi-symmetric functions is commonly called
or occasionally just
(when
is clear from the context or
or
). It is graded by
the total degree of the power series. Its homogeneous elements of degree
form a free
-submodule of rank equal to the number of
compositions of
(that is,
if
, else
).
The two classical bases of , the monomial basis
and the fundamental basis
, are indexed by
compositions
and defined by the
formulas:
and
where in the second equation the sum runs over all weakly increasing
-tuples
of positive integers
(where
is the size of
) which increase strictly from
to
if
is a descent of the composition
.
These bases are related by the formula
where the inequality indicates that
is finer than
.
The -algebra of quasi-symmetric functions is a Hopf algebra,
with the coproduct satisfying
for every composition .
It is possible to define an -algebra of quasi-symmetric
functions in a finite number of variables as well (but it is not
a bialgebra). These quasi-symmetric functions are actual polynomials
then, not just power series.
Chapter 5 of [Reiner2013] and Section 11 of [HazWitt1] are devoted to quasi-symmetric functions, as are Malvenuto’s thesis [Mal1993] and part of Chapter 7 of [Sta1999].
The implementation of the quasi-symmetric function Hopf algebra
We realize the -algebra of quasi-symmetric functions in Sage as
a graded Hopf algebra with basis elements indexed by compositions.
sage: QSym = QuasiSymmetricFunctions(QQ)
sage: QSym.category()
Join of Category of graded hopf algebras over Rational Field and Category of monoids with realizations and Category of coalgebras over Rational Field with realizations
The most standard two bases for this -algebra are the monomial and
fundamental bases, and are accessible by the M() and F() methods:
sage: M = QSym.M()
sage: F = QSym.F()
sage: M(F[2,1,2])
M[1, 1, 1, 1, 1] + M[1, 1, 1, 2] + M[2, 1, 1, 1] + M[2, 1, 2]
sage: F(M[2,1,2])
F[1, 1, 1, 1, 1] - F[1, 1, 1, 2] - F[2, 1, 1, 1] + F[2, 1, 2]
The product on this space is commutative and is inherited from the product on the realization within the ring of power series:
sage: M[3]*M[1,1] == M[1,1]*M[3]
True
sage: M[3]*M[1,1]
M[1, 1, 3] + M[1, 3, 1] + M[1, 4] + M[3, 1, 1] + M[4, 1]
sage: F[3]*F[1,1]
F[1, 1, 3] + F[1, 2, 2] + F[1, 3, 1] + F[1, 4] + F[2, 1, 2] + F[2, 2, 1] + F[2, 3] + F[3, 1, 1] + F[3, 2] + F[4, 1]
sage: M[3]*F[2]
M[1, 1, 3] + M[1, 3, 1] + M[1, 4] + M[2, 3] + M[3, 1, 1] + M[3, 2] + M[4, 1] + M[5]
sage: F[2]*M[3]
F[1, 1, 1, 2] - F[1, 2, 2] + F[2, 1, 1, 1] - F[2, 1, 2] - F[2, 2, 1] + F[5]
There is a coproduct on as well, which in the Monomial
basis acts by cutting the composition into a left half and a right
half. The coproduct is not co-commutative:
sage: M[1,3,1].coproduct()
M[] # M[1, 3, 1] + M[1] # M[3, 1] + M[1, 3] # M[1] + M[1, 3, 1] # M[]
sage: F[1,3,1].coproduct()
F[] # F[1, 3, 1] + F[1] # F[3, 1] + F[1, 1] # F[2, 1] + F[1, 2] # F[1, 1] + F[1, 3] # F[1] + F[1, 3, 1] # F[]
The duality pairing with non-commutative symmetric functions
These two operations endow the quasi-symmetric functions
with the structure of a Hopf algebra. It is the graded
dual Hopf algebra of the non-commutative symmetric functions
.
Under this duality, the Monomial basis of
is dual to
the Complete basis of
, and the Fundamental basis of
is dual to the Ribbon basis of
(see [MR]).
sage: S = M.dual(); S
Non-Commutative Symmetric Functions over the Rational Field in the Complete basis
sage: M[1,3,1].duality_pairing( S[1,3,1] )
1
sage: M.duality_pairing_matrix( S, degree=4 )
[1 0 0 0 0 0 0 0]
[0 1 0 0 0 0 0 0]
[0 0 1 0 0 0 0 0]
[0 0 0 1 0 0 0 0]
[0 0 0 0 1 0 0 0]
[0 0 0 0 0 1 0 0]
[0 0 0 0 0 0 1 0]
[0 0 0 0 0 0 0 1]
sage: F.duality_pairing_matrix( S, degree=4 )
[1 0 0 0 0 0 0 0]
[1 1 0 0 0 0 0 0]
[1 0 1 0 0 0 0 0]
[1 1 1 1 0 0 0 0]
[1 0 0 0 1 0 0 0]
[1 1 0 0 1 1 0 0]
[1 0 1 0 1 0 1 0]
[1 1 1 1 1 1 1 1]
sage: NCSF = M.realization_of().dual()
sage: R = NCSF.Ribbon()
sage: F.duality_pairing_matrix( R, degree=4 )
[1 0 0 0 0 0 0 0]
[0 1 0 0 0 0 0 0]
[0 0 1 0 0 0 0 0]
[0 0 0 1 0 0 0 0]
[0 0 0 0 1 0 0 0]
[0 0 0 0 0 1 0 0]
[0 0 0 0 0 0 1 0]
[0 0 0 0 0 0 0 1]
sage: M.duality_pairing_matrix( R, degree=4 )
[ 1 0 0 0 0 0 0 0]
[-1 1 0 0 0 0 0 0]
[-1 0 1 0 0 0 0 0]
[ 1 -1 -1 1 0 0 0 0]
[-1 0 0 0 1 0 0 0]
[ 1 -1 0 0 -1 1 0 0]
[ 1 0 -1 0 -1 0 1 0]
[-1 1 1 -1 1 -1 -1 1]
Let and
be elements of
, and
an element of
. Then, if we represent the duality pairing with the
mathematical notation
,
For example, the coefficient of M[2,1,4,1] in M[1,3]*M[2,1,1] may be computed with the duality pairing:
sage: I, J = Composition([1,3]), Composition([2,1,1])
sage: (M[I]*M[J]).duality_pairing(S[2,1,4,1])
1
And the coefficient of S[1,3] # S[2,1,1] in S[2,1,4,1].coproduct() is equal to this result:
sage: S[2,1,4,1].coproduct()
S[] # S[2, 1, 4, 1] + ... + S[1, 3] # S[2, 1, 1] + ... + S[4, 1] # S[2, 1]
The duality pairing on the tensor space is another way of getting this coefficient, but currently the method duality_pairing is not defined on the tensor squared space. However, we can extend this functionality by applying a linear morphism to the terms in the coproduct, as follows:
sage: X = S[2,1,4,1].coproduct()
sage: def linear_morphism(x, y):
....: return x.duality_pairing(M[1,3]) * y.duality_pairing(M[2,1,1])
sage: X.apply_multilinear_morphism(linear_morphism, codomain=ZZ)
1
Similarly, if is an element of
and
and
are
elements of
, then
For example, the coefficient of R[2,3,1] in R[2,1]*R[2,1] is computed with the duality pairing by the following command:
sage: (R[2,1]*R[2,1]).duality_pairing(F[2,3,1])
1
sage: R[2,1]*R[2,1]
R[2, 1, 2, 1] + R[2, 3, 1]
This coefficient should then be equal to the coefficient of F[2,1] # F[2,1] in F[2,3,1].coproduct():
sage: F[2,3,1].coproduct()
F[] # F[2, 3, 1] + ... + F[2, 1] # F[2, 1] + ... + F[2, 3, 1] # F[]
This can also be computed by the duality pairing on the tensor space, as above:
sage: X = F[2,3,1].coproduct()
sage: def linear_morphism(x, y):
....: return x.duality_pairing(R[2,1]) * y.duality_pairing(R[2,1])
sage: X.apply_multilinear_morphism(linear_morphism, codomain=ZZ)
1
The operation dual to multiplication by a non-commutative symmetric function
Let and consider the linear endomorphism of
defined by
left (respectively, right) multiplication by
. Since there is a duality
between
and
, this linear transformation induces an
operator
on
satisfying
for any non-commutative symmetric function .
This is implemented by the method skew_by(). Explicitly, if H is a quasi-symmetric function and g a non-commutative symmetric function, then H.skew_by(g) and H.skew_by(g, side='right') are expressions that satisfy, for any non-commutative symmetric function h, the following equalities:
H.skew_by(g).duality_pairing(h) == H.duality_pairing(g*h)
H.skew_by(g, side='right').duality_pairing(h) == H.duality_pairing(h*g)
For example, M[J].skew_by(S[I]) is unless the composition J
begins with I and M(J).skew_by(S(I), side='right') is
unless
the composition J ends with I. For example:
sage: M[3,2,2].skew_by(S[3])
M[2, 2]
sage: M[3,2,2].skew_by(S[2])
0
sage: M[3,2,2].coproduct().apply_multilinear_morphism( lambda x,y: x.duality_pairing(S[3])*y )
M[2, 2]
sage: M[3,2,2].skew_by(S[3], side='right')
0
sage: M[3,2,2].skew_by(S[2], side='right')
M[3, 2]
The counit
The counit is defined by sending all elements of positive degree to zero:
sage: M[3].degree(), M[3,1,2].degree(), M.one().degree()
(3, 6, 0)
sage: M[3].counit()
0
sage: M[3,1,2].counit()
0
sage: M.one().counit()
1
sage: (M[3] - 2*M[3,1,2] + 7).counit()
7
sage: (F[3] - 2*F[3,1,2] + 7).counit()
7
The antipode
The antipode sends the Fundamental basis element indexed by the
composition to
to the size of
times the Fundamental
basis element indexed by the conjugate composition to
.
sage: F[3,2,2].antipode()
-F[1, 2, 2, 1, 1]
sage: Composition([3,2,2]).conjugate()
[1, 2, 2, 1, 1]
sage: M[3,2,2].antipode()
-M[2, 2, 3] - M[2, 5] - M[4, 3] - M[7]
We demonstrate here the defining relation of the antipode:
sage: X = F[3,2,2].coproduct()
sage: X.apply_multilinear_morphism(lambda x,y: x*y.antipode())
0
sage: X.apply_multilinear_morphism(lambda x,y: x.antipode()*y)
0
The relation with symmetric functions
The quasi-symmetric functions are a ring which contain the symmetric functions as a subring. The Monomial quasi-symmetric functions are related to the monomial symmetric functions by
There are methods to test if an expression in the quasi-symmetric functions is a symmetric function and, if it is, send it to an expression in the symmetric functions:
sage: f = M[1,1,2] + M[1,2,1]
sage: f.is_symmetric()
False
sage: g = M[3,1] + M[1,3]
sage: g.is_symmetric()
True
sage: g.to_symmetric_function()
m[3, 1]
The expansion of the Schur function in terms of the Fundamental quasi-symmetric functions is due to [Ges]. There is one term in the expansion for each standard tableau of shape equal to the partition indexing the Schur function.
sage: f = F[3,2] + F[2,2,1] + F[2,3] + F[1,3,1] + F[1,2,2]
sage: f.is_symmetric()
True
sage: f.to_symmetric_function()
5*m[1, 1, 1, 1, 1] + 3*m[2, 1, 1, 1] + 2*m[2, 2, 1] + m[3, 1, 1] + m[3, 2]
sage: s = SymmetricFunctions(QQ).s()
sage: s(f.to_symmetric_function())
s[3, 2]
It is also possible to convert a symmetric function to a quasi-symmetric function:
sage: m = SymmetricFunctions(QQ).m()
sage: M( m[3,1,1] )
M[1, 1, 3] + M[1, 3, 1] + M[3, 1, 1]
sage: F( s[2,2,1] )
F[1, 1, 2, 1] + F[1, 2, 1, 1] + F[1, 2, 2] + F[2, 1, 2] + F[2, 2, 1]
It is possible to experiment with the quasi-symmetric function expansion of other bases, but it is important that the base ring be the same for both algebras.
sage: R = QQ['t']
sage: Qp = SymmetricFunctions(R).hall_littlewood().Qp()
sage: QSymt = QuasiSymmetricFunctions(R)
sage: Ft = QSymt.F()
sage: Ft( Qp[2,2] )
F[1, 2, 1] + t*F[1, 3] + (t+1)*F[2, 2] + t*F[3, 1] + t^2*F[4]
sage: K = QQ['q','t'].fraction_field()
sage: Ht = SymmetricFunctions(K).macdonald().Ht()
sage: Fqt = QuasiSymmetricFunctions(Ht.base_ring()).F()
sage: Fqt(Ht[2,1])
q*t*F[1, 1, 1] + (q+t)*F[1, 2] + (q+t)*F[2, 1] + F[3]
The following will raise an error because the base ring of F is not equal to the base ring of Ht:
sage: F(Ht[2,1])
Traceback (most recent call last):
...
TypeError: do not know how to make x (= McdHt[2, 1]) an element of self (=Quasisymmetric functions over the Rational Field in the Fundamental basis)
The map to the ring of polynomials
The quasi-symmetric functions can be seen as an inverse limit
of a subring of a polynomial ring as the number of variables
increases. Indeed, there exists a projection from the
quasi-symmetric functions onto the polynomial ring
. This projection is defined by
sending the variables
to
, while
the remaining
variables remain fixed. Note that this
projection sends
to
if the length of the composition
is higher than
.
sage: M[1,3,1].expand(4)
x0*x1^3*x2 + x0*x1^3*x3 + x0*x2^3*x3 + x1*x2^3*x3
sage: F[1,3,1].expand(4)
x0*x1^3*x2 + x0*x1^3*x3 + x0*x1^2*x2*x3 + x0*x1*x2^2*x3 + x0*x2^3*x3 + x1*x2^3*x3
sage: M[1,3,1].expand(2)
0
TESTS:
sage: QSym = QuasiSymmetricFunctions(QQ); QSym
Quasisymmetric functions over the Rational Field
sage: QSym.base_ring()
Rational Field
Bases: sage.categories.realizations.Category_realization_of_parent
Category of bases of quasi-symmetric functions.
EXAMPLES:
sage: QSym = QuasiSymmetricFunctions(QQ)
sage: QSym.Bases()
Category of bases of Quasisymmetric functions over the Rational Field
alias of Bases.ElementMethods
alias of Bases.ParentMethods
Return the super categories of bases of the Quasi-symmetric functions.
OUTPUT:
TESTS:
sage: QSym = QuasiSymmetricFunctions(QQ)
sage: QSym.Bases().super_categories()
[Category of bases of Non-Commutative Symmetric Functions or Quasisymmetric functions over the Rational Field, Category of commutative rings]
Bases: sage.combinat.free_module.CombinatorialFreeModule, sage.misc.bindable_class.BindableClass
The Hopf algebra of quasi-symmetric functions in the Fundamental basis.
EXAMPLES:
sage: QSym = QuasiSymmetricFunctions(QQ)
sage: F = QSym.F()
sage: M = QSym.M()
sage: F(M[2,2])
F[1, 1, 1, 1] - F[1, 1, 2] - F[2, 1, 1] + F[2, 2]
sage: s = SymmetricFunctions(QQ).s()
sage: F(s[3,2])
F[1, 2, 2] + F[1, 3, 1] + F[2, 2, 1] + F[2, 3] + F[3, 2]
sage: (1+F[1])^3
F[] + 3*F[1] + 3*F[1, 1] + F[1, 1, 1] + 2*F[1, 2] + 3*F[2] + 2*F[2, 1] + F[3]
sage: F[1,2,1].coproduct()
F[] # F[1, 2, 1] + F[1] # F[2, 1] + F[1, 1] # F[1, 1] + F[1, 2] # F[1] + F[1, 2, 1] # F[]
The following is an alias for this basis:
sage: QSym.Fundamental()
Quasisymmetric functions over the Rational Field in the Fundamental basis
TESTS:
sage: F(M([]))
F[]
sage: F(M(0))
0
sage: F(s([]))
F[]
sage: F(s(0))
0
alias of Fundamental.Element
Return the antipode to a Fundamental quasi-symmetric basis element.
INPUT:
OUTPUT:
EXAMPLES:
sage: F = QuasiSymmetricFunctions(QQ).F()
sage: F.antipode_on_basis(Composition([2,1]))
-F[2, 1]
Return the coproduct to a Fundamental quasi-symmetric basis element.
Combinatorial rule: quasi-deconcatenation.
INPUT:
OUTPUT:
EXAMPLES:
sage: F = QuasiSymmetricFunctions(QQ).Fundamental()
sage: F[4].coproduct()
F[] # F[4] + F[1] # F[3] + F[2] # F[2] + F[3] # F[1] + F[4] # F[]
sage: F[2,1,3].coproduct()
F[] # F[2, 1, 3] + F[1] # F[1, 1, 3] + F[2] # F[1, 3] + F[2, 1] # F[3] + F[2, 1, 1] # F[2] + F[2, 1, 2] # F[1] + F[2, 1, 3] # F[]
TESTS:
sage: F.coproduct_on_basis(Composition([2,1,3]))
F[] # F[2, 1, 3] + F[1] # F[1, 1, 3] + F[2] # F[1, 3] + F[2, 1] # F[3] + F[2, 1, 1] # F[2] + F[2, 1, 2] # F[1] + F[2, 1, 3] # F[]
sage: F.one().coproduct() # generic for graded / graded connected
F[] # F[]
Return the dual basis to the Fundamental basis. This is the ribbon basis of the non-commutative symmetric functions.
OUTPUT:
EXAMPLES:
sage: F = QuasiSymmetricFunctions(QQ).F()
sage: F.dual()
Non-Commutative Symmetric Functions over the Rational Field in the Ribbon basis
Bases: sage.combinat.free_module.CombinatorialFreeModule, sage.misc.bindable_class.BindableClass
The Hazewinkel lambda basis of the quasi-symmetric functions.
This basis goes back to [Haz2004], albeit it is indexed in a different way here. It is a multiplicative basis in a weak sense of this word (the product of any two basis elements is a basis element, but of course not the one obtained by concatenating the indexing compositions).
In [Haz2004], Hazewinkel showed that the -algebra
is a polynomial algebra. (The proof is correct
but rests upon an unproven claim that the lexicographically
largest term of the
-th shuffle power of a Lyndon word is
the
-fold concatenatenation of this Lyndon word with
itself, occuring
times in that shuffle power. But this
can be deduced from Section 2 of [Rad1979].) More precisely,
he showed that
is generated, as a free
commutative
-algebra, by the elements
, where
ranges over the positive integers,
and
ranges over all compositions which are Lyndon words
and whose entries have gcd
. Here,
denotes the
-th lambda operation as explained in
lambda_of_monomial().
Thus, products of these generators form a -module
basis of
. We index this basis by compositions
here. More precisely, we define the Hazewinkel lambda basis
(with
ranging over all compositions)
as follows:
Let be a composition. Let
be the
Chen-Fox-Lyndon factorization of
(see
lyndon_factorization()
). For every
, let
be the
gcd of the entries of the Lyndon word
, and let
be
the result of dividing the entries of
by this gcd. Then,
is defined to be
.
Todo
The conversion from the M basis to the HWL basis is currently implemented in the naive way (inverting the base-change matrix in the other direction). This matrix is not triangular (not even after any permutations of the bases), and there could very well be a faster method (the one given by Hazewinkel?).
EXAMPLES:
sage: QSym = QuasiSymmetricFunctions(ZZ)
sage: HWL = QSym.HazewinkelLambda()
sage: M = QSym.M()
sage: M(HWL([2]))
M[1, 1]
sage: M(HWL([1,1]))
2*M[1, 1] + M[2]
sage: M(HWL([1,2]))
M[1, 2]
sage: M(HWL([2,1]))
3*M[1, 1, 1] + M[1, 2] + M[2, 1]
sage: M(HWL(Composition([])))
M[]
sage: HWL(M([1,1]))
HWL[2]
sage: HWL(M(Composition([2])))
HWL[1, 1] - 2*HWL[2]
sage: HWL(M([1]))
HWL[1]
TESTS:
Transforming from the M-basis into the HWL-basis and back returns us to where we started:
sage: all( M(HWL(M[I])) == M[I] for I in Compositions(3) )
True
sage: all( HWL(M(HWL[I])) == HWL[I] for I in Compositions(4) )
True
Checking the HWL basis elements corresponding to Lyndon words:
sage: all( M(HWL[Composition(I)])
....: == M.lambda_of_monomial([i // gcd(I) for i in I], gcd(I))
....: for I in LyndonWords(e=3, k=2) )
True
The product on Hazewinkel Lambda basis elements.
The product of the basis elements indexed by two compositions
and
is the basis element obtained by concatenating the
Lyndon factorizations of the words
and
, then reordering
the Lyndon factors in nonincreasing order, and finally
concatenating them in this order (giving a new composition).
INPUT:
OUTPUT:
EXAMPLES:
sage: HWL = QuasiSymmetricFunctions(QQ).HazewinkelLambda()
sage: c1 = Composition([1, 2, 1])
sage: c2 = Composition([2, 1, 3, 2])
sage: HWL.product_on_basis(c1, c2)
HWL[2, 1, 3, 2, 1, 2, 1]
sage: HWL.product_on_basis(c1, Composition([]))
HWL[1, 2, 1]
sage: HWL.product_on_basis(Composition([]), Composition([]))
HWL[]
TESTS:
sage: M = QuasiSymmetricFunctions(QQ).M()
sage: all( all( M(HWL[I] * HWL[J]) == M(HWL[I]) * M(HWL[J])
....: for I in Compositions(3) )
....: for J in Compositions(3) )
True
Bases: sage.combinat.free_module.CombinatorialFreeModule, sage.misc.bindable_class.BindableClass
The Hopf algebra of quasi-symmetric function in the Monomial basis.
EXAMPLES:
sage: QSym = QuasiSymmetricFunctions(QQ)
sage: M = QSym.M()
sage: F = QSym.F()
sage: M(F[2,2])
M[1, 1, 1, 1] + M[1, 1, 2] + M[2, 1, 1] + M[2, 2]
sage: m = SymmetricFunctions(QQ).m()
sage: M(m[3,1,1])
M[1, 1, 3] + M[1, 3, 1] + M[3, 1, 1]
sage: (1+M[1])^3
M[] + 3*M[1] + 6*M[1, 1] + 6*M[1, 1, 1] + 3*M[1, 2] + 3*M[2] + 3*M[2, 1] + M[3]
sage: M[1,2,1].coproduct()
M[] # M[1, 2, 1] + M[1] # M[2, 1] + M[1, 2] # M[1] + M[1, 2, 1] # M[]
The following is an alias for this basis:
sage: QSym.Monomial()
Quasisymmetric functions over the Rational Field in the Monomial basis
TESTS:
sage: M(F([]))
M[]
sage: M(F(0))
0
sage: M(m([]))
M[]
alias of Monomial.Element
Return the result of the antipode applied to a quasi-symmetric Monomial basis element.
INPUT:
OUTPUT:
EXAMPLES:
sage: M = QuasiSymmetricFunctions(QQ).M()
sage: M.antipode_on_basis(Composition([2,1]))
M[1, 2] + M[3]
sage: M.antipode_on_basis(Composition([]))
M[]
Return the coproduct of a Monomial basis element.
Combinatorial rule: deconcatenation.
INPUT:
OUTPUT:
EXAMPLES:
sage: M=QuasiSymmetricFunctions(QQ).Monomial()
sage: M[4,2,3].coproduct()
M[] # M[4, 2, 3] + M[4] # M[2, 3] + M[4, 2] # M[3] + M[4, 2, 3] # M[]
sage: M.coproduct_on_basis(Composition([]))
M[] # M[]
Return the dual basis to the Monomial basis. This is the complete basis of the non-commutative symmetric functions.
OUTPUT:
EXAMPLES:
sage: M = QuasiSymmetricFunctions(QQ).M()
sage: M.dual()
Non-Commutative Symmetric Functions over the Rational Field in the Complete basis
Return the image of the monomial quasi-symmetric function
under the lambda-map
, expanded in the
monomial basis.
The ring of quasi-symmetric functions over the integers,
(and more generally, the ring of
quasi-symmetric functions over any binomial ring) becomes
a
-ring (with the
-structure inherited
from the ring of formal power series, so that
is
if
and
if
).
The Adams operations of this -ring are the
Frobenius endomorphisms
(see
frobenius()
for their definition). Using these endomorphisms, the
-operations can be explicitly computed via the formula
in the ring of formal power series in a variable over
the ring of quasi-symmetric functions. In particular,
every composition
satisfies
(corrected version of Remark 2.4 in [Haz2004]).
The quasi-symmetric functions with
ranging over the positive integers and
ranging over
the reduced Lyndon compositions (i. e., compositions
which are Lyndon words and have the gcd of their entries
equal to
) form a set of free polynomial generators
for
. See [Haz2004] for a major part
of the proof.
INPUT:
OUTPUT:
The quasi-symmetric function , expanded in
the monomial basis over the ground ring of self.
EXAMPLES:
sage: M = QuasiSymmetricFunctions(CyclotomicField()).Monomial()
sage: M.lambda_of_monomial([1, 2], 2)
2*M[1, 1, 2, 2] + M[1, 1, 4] + M[1, 2, 1, 2] + M[1, 3, 2] + M[2, 2, 2]
sage: M.lambda_of_monomial([1, 1], 2)
3*M[1, 1, 1, 1] + M[1, 1, 2] + M[1, 2, 1] + M[2, 1, 1]
sage: M = QuasiSymmetricFunctions(Integers(19)).Monomial()
sage: M.lambda_of_monomial([1, 2], 3)
6*M[1, 1, 1, 2, 2, 2] + 3*M[1, 1, 1, 2, 4] + 3*M[1, 1, 1, 4, 2]
+ M[1, 1, 1, 6] + 4*M[1, 1, 2, 1, 2, 2] + 2*M[1, 1, 2, 1, 4]
+ 2*M[1, 1, 2, 2, 1, 2] + 2*M[1, 1, 2, 3, 2] + 4*M[1, 1, 3, 2, 2]
+ 2*M[1, 1, 3, 4] + M[1, 1, 4, 1, 2] + M[1, 1, 5, 2]
+ 2*M[1, 2, 1, 1, 2, 2] + M[1, 2, 1, 1, 4] + M[1, 2, 1, 2, 1, 2]
+ M[1, 2, 1, 3, 2] + 4*M[1, 2, 2, 2, 2] + M[1, 2, 2, 4] + M[1, 2, 4, 2]
+ 2*M[1, 3, 1, 2, 2] + M[1, 3, 1, 4] + M[1, 3, 2, 1, 2] + M[1, 3, 3, 2]
+ M[1, 4, 2, 2] + 3*M[2, 1, 2, 2, 2] + M[2, 1, 2, 4] + M[2, 1, 4, 2]
+ 2*M[2, 2, 1, 2, 2] + M[2, 2, 1, 4] + M[2, 2, 2, 1, 2] + M[2, 2, 3, 2]
+ 2*M[2, 3, 2, 2] + M[2, 3, 4] + M[3, 2, 2, 2]
The map sends everything to
:
sage: M = QuasiSymmetricFunctions(ZZ).Monomial()
sage: all( M.lambda_of_monomial(I, 0) == M.one()
....: for I in Compositions(3) )
True
The map is the identity map:
sage: M = QuasiSymmetricFunctions(QQ).Monomial()
sage: all( M.lambda_of_monomial(I, 1) == M(I)
....: for I in Compositions(3) )
True
sage: M = QuasiSymmetricFunctions(Integers(5)).Monomial()
sage: all( M.lambda_of_monomial(I, 1) == M(I)
....: for I in Compositions(3) )
True
sage: M = QuasiSymmetricFunctions(ZZ).Monomial()
sage: all( M.lambda_of_monomial(I, 1) == M(I)
....: for I in Compositions(3) )
True
The product on Monomial basis elements.
The product of the basis elements indexed by two compositions
and
is the sum of the basis elements indexed by
compositions in the stuffle product (also called the
overlapping shuffle product) of
and
.
INPUT:
OUTPUT:
EXAMPLES:
sage: M = QuasiSymmetricFunctions(QQ).Monomial()
sage: c1 = Composition([2])
sage: c2 = Composition([1,3])
sage: M.product_on_basis(c1, c2)
M[1, 2, 3] + M[1, 3, 2] + M[1, 5] + M[2, 1, 3] + M[3, 3]
sage: M.product_on_basis(c1, Composition([]))
M[2]
Bases: sage.combinat.free_module.CombinatorialFreeModule, sage.misc.bindable_class.BindableClass
The Hopf algebra of quasi-symmetric function in the Quasisymmetric Schur basis.
The basis of Quasisymmetric Schur functions is defined in [QSCHUR].
EXAMPLES:
sage: QSym = QuasiSymmetricFunctions(QQ)
sage: QS = QSym.QS()
sage: F = QSym.F()
sage: M = QSym.M()
sage: F(QS[1,2])
F[1, 2]
sage: M(QS[1,2])
M[1, 1, 1] + M[1, 2]
sage: s = SymmetricFunctions(QQ).s()
sage: QS(s[2,1,1])
QS[1, 1, 2] + QS[1, 2, 1] + QS[2, 1, 1]
Return the realization of the Monomial basis of the ring of quasi-symmetric functions.
OUTPUT:
EXAMPLES:
sage: QuasiSymmetricFunctions(QQ).a_realization()
Quasisymmetric functions over the Rational Field in the Monomial basis
Return the dual Hopf algebra of the quasi-symmetric functions, which is the non-commutative symmetric functions.
OUTPUT:
EXAMPLES:
sage: QSym = QuasiSymmetricFunctions(QQ)
sage: QSym.dual()
Non-Commutative Symmetric Functions over the Rational Field
Bases: sage.combinat.free_module.CombinatorialFreeModule, sage.misc.bindable_class.BindableClass
The dual immaculate basis of the quasi-symmetric functions.
This basis first appears in [BBSSZ2012].
REFERENCES:
[BBSSZ2012] | Chris Berg, Nantel Bergeron, Franco Saliola, Luis Serrano, Mike Zabrocki, A lift of the Schur and Hall-Littlewood bases to non-commutative symmetric functions, Arxiv 1208.5191v3. |
EXAMPLES:
sage: QSym = QuasiSymmetricFunctions(QQ)
sage: dI = QSym.dI()
sage: dI([1,3,2])*dI([1]) # long time (6s on sage.math, 2013)
dI[1, 1, 3, 2] + dI[2, 3, 2]
sage: dI([1,3])*dI([1,1])
dI[1, 1, 1, 3] + dI[1, 1, 4] + dI[1, 2, 3] - dI[1, 3, 2] - dI[1, 4, 1] - dI[1, 5] + dI[2, 3, 1] + dI[2, 4]
sage: dI([3,1])*dI([2,1]) # long time (7s on sage.math, 2013)
dI[1, 1, 5] - dI[1, 4, 1, 1] - dI[1, 4, 2] - 2*dI[1, 5, 1] - dI[1, 6] - dI[2, 4, 1] - dI[2, 5] - dI[3, 1, 3] + dI[3, 2, 1, 1] + dI[3, 2, 2] + dI[3, 3, 1] + dI[4, 1, 1, 1] + 2*dI[4, 2, 1] + dI[4, 3] + dI[5, 1, 1] + dI[5, 2]
sage: F = QSym.F()
sage: dI(F[1,3,1])
-dI[1, 1, 1, 2] + dI[1, 1, 2, 1] - dI[1, 2, 2] + dI[1, 3, 1]
sage: F(dI(F([2,1,3])))
F[2, 1, 3]
Return the quasi-symmetric function in the Monomial basis corresponding to the quasi-symmetric polynomial f.
INPUT:
OUTPUT:
EXAMPLES:
sage: P = PolynomialRing(QQ, 'x', 3)
sage: x = P.gens()
sage: f = x[0] + x[1] + x[2]
sage: QSym = QuasiSymmetricFunctions(QQ)
sage: QSym.from_polynomial(f)
M[1]
Beware of setting check=False:
sage: f = x[0] + 2*x[1] + x[2]
sage: QSym.from_polynomial(f, check=True)
Traceback (most recent call last):
...
ValueError: x0 + 2*x1 + x2 is not a quasi-symmetric polynomial
sage: QSym.from_polynomial(f, check=False)
M[1]
To expand the quasi-symmetric function in a basis other than the Monomial basis, the following shorthands are provided:
sage: M = QSym.Monomial()
sage: f = x[0]**2+x[1]**2+x[2]**2
sage: g = M.from_polynomial(f); g
M[2]
sage: F = QSym.Fundamental()
sage: F(g)
-F[1, 1] + F[2]
sage: F.from_polynomial(f)
-F[1, 1] + F[2]