In this document we briefly explain the quasisymmetric function bases and related functionality in Sage. We assume the reader is familar with the package SymmetricFunctions.
Quasisymmetric functions, denoted , form a subring of the power
series ring in countably many variables.
contains the symmetric
functions. These functions first arose in the theory of
-partitions. The initial ideas in this field are attributed to
MacMahon, Knuth, Kreweras, Glâffrwd Thomas, Stanley. In 1984, Gessel
formalized the study of quasisymmetric functions and introduced the
basis of fundamental quasisymmetric functions [Ges]. In 1995, Gelfand,
Krob, Lascoux, Leclerc, Retakh, and Thibon showed that the ring of
quasisymmetric functions is Hopf dual to the noncommutative symmetric
functions [NCSF]. Many results have built on these.
One advantage of working in is that many interesting families of
symmetric functions have explicit expansions in fundamental quasisymmetric
functions such as Schur functions [Ges], Macdonald polynomials
[HHL05], and plethysm of Schur functions [LW12].
For more background see Wikipedia article Quasisymmetric_function.
To begin, initialize the ring. Below we chose to use the rational
numbers . Other options include the integers
and
:
sage: QSym = QuasiSymmetricFunctions(QQ)
sage: QSym
Quasisymmetric functions over the Rational Field
sage: QSym = QuasiSymmetricFunctions(CC); QSym
Quasisymmetric functions over the Complex Field with 53 bits of precision
sage: QSym = QuasiSymmetricFunctions(ZZ); QSym
Quasisymmetric functions over the Integer Ring
All bases of are indexed by compositions e.g.
. The
convention is to use capitol letters for bases of
and lowercase
letters for bases of the symmetric functions
. Next set up names for the
known bases by running inject_shorthands(). As with symmetric functions,
you do not need to run this commmand and you could assign these bases other
names.
sage: QSym = QuasiSymmetricFunctions(QQ)
sage: QSym.inject_shorthands()
Injecting M as shorthand for Quasisymmetric functions over the Rational Field in the Monomial basis
Injecting F as shorthand for Quasisymmetric functions over the Rational Field in the Fundamental basis
Injecting dI as shorthand for Quasisymmetric functions over the Rational Field in the dualImmaculate basis
Now one can start constructing quasisymmetric functions.
Note
It is best to use variables other than M and F.
sage: x = M[2,1] + M[1,2]
sage: x
M[1, 2] + M[2, 1]
sage: y = 3*M[1,2] + M[3]^2; y
3*M[1, 2] + 2*M[3, 3] + M[6]
sage: F[3,1,3] + 7*F[2,1]
7*F[2, 1] + F[3, 1, 3]
sage: 3*F[2,1,2] + F[3]^2
F[1, 2, 2, 1] + F[1, 2, 3] + 2*F[1, 3, 2] + F[1, 4, 1] + F[1, 5] + 3*F[2, 1, 2]
+ 2*F[2, 2, 2] + 2*F[2, 3, 1] + 2*F[2, 4] + F[3, 2, 1] + 3*F[3, 3] + 2*F[4, 2] + F[5, 1] + F[6]
To convert from one basis to another is easy:
sage: z = M[1,2,1]
sage: z
M[1, 2, 1]
sage: F(z)
-F[1, 1, 1, 1] + F[1, 2, 1]
sage: M(F(z))
M[1, 2, 1]
To expand in variables, one can specify a finite size alphabet :
sage: y = M[1,2,1]
sage: y.expand(4)
x0*x1^2*x2 + x0*x1^2*x3 + x0*x2^2*x3 + x1*x2^2*x3
The usual methods on free modules are available such as coefficients, degrees, and the support:
sage: z=3*M[1,2]+M[3]^2; z
3*M[1, 2] + 2*M[3, 3] + M[6]
sage: z.coefficient([1,2])
3
sage: z.degree()
6
sage: z.coefficients()
[3, 2, 1]
sage: z.monomials()
[M[1, 2], M[3, 3], M[6]]
sage: z.monomial_coefficients()
{[3, 3]: 2, [1, 2]: 3, [6]: 1}
As with the symmetric functions package, the quasisymmetric function 1 has several instantiations. However, the most obvious way to write 1 leads to an error (this is due to the semantics of python):
sage: M[[]]
M[]
sage: M.one()
M[]
sage: M(1)
M[]
sage: M[[]] == 1
True
sage: M[]
Traceback (most recent call last):
...
SyntaxError: invalid syntax
The quasisymmetric functions are a ring which contains the symmetric
functions as a subring. The Monomial quasisymmetric functions are
related to the monomial symmetric functions by :
sage: SymmetricFunctions(QQ).inject_shorthands()
doctest:...: RuntimeWarning: redefining global value `e`
sage: m[2,1]
m[2, 1]
sage: M(m[2,1])
M[1, 2] + M[2, 1]
sage: M(s[2,1])
2*M[1, 1, 1] + M[1, 2] + M[2, 1]
There are methods to test if an expression in the quasisymmetric functions
is a symmetric function:
sage: f = M[1,1,2] + M[1,2,1]
sage: f.is_symmetric()
False
sage: f = M[3,1] + M[1,3]
sage: f.is_symmetric()
True
If is symmetric, there are methods to convert
to an expression in the
symmetric functions:
sage: f.to_symmetric_function()
m[3, 1]
The expansion of the Schur function in terms of the Fundamental quasisymmetric 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(f.to_symmetric_function())
s[3, 2]
It is also possible to convert any symmetric function to the quasisymmetric function expansion in any known basis. The converse is not true:
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]
sage: s(M[2,1])
Traceback (most recent call last):
...
TypeError: do not know how to make x (= M[2, 1]) an element of self
It is possible to experiment with the quasisymmetric 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 product on this space is commutative and is inherited from the product by the realization within the polynomial ring:
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 this ring as well, which in the Monomial basis acts by cutting the composition into a left half and a right half. The co-product is non-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 with the structure of a Hopf algebra. It is
the 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
, we have:
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 Adjoint 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 quasisymmetric 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.
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
begins with
and M(J).skew_by(S(I), side='right') is
unless
the composition
ends with
:
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 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
REFERENCES:
[HHL05] | A combinatorial formula for Macdonald polynomials. Haiman, Haglund, and Loehr. J. Amer. Math. Soc. 18 (2005), no. 3, 735-761. |
[LW12] | Quasisymmetric expansions of Schur-function plethysms. Loehr and Warrington. Proc. Amer. Math. Soc. 140 (2012), no. 4, 1159-1171. |
[KT97] | Noncommutative symmetric functions IV: Quantum linear groups and
Hecke algebras at ![]() |