Introduction to Quasisymmetric Functions

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 \(QSym\), form a subring of the power series ring in countably many variables. \(QSym\) contains the symmetric functions. These functions first arose in the theory of \(P\)-partitions. The initial ideas in this field are attributed to MacMahon, Knuth, Kreweras, Glânffrwd 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 \(QSym\) 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 \(\QQ\). Other options include the integers \(\ZZ\) and \(\CC\):

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 \(QSym\) are indexed by compositions e.g. \([3,1,1,4]\). The convention is to use capital letters for bases of \(QSym\) and lowercase letters for bases of the symmetric functions \(Sym\). Next set up names for the known bases by running inject_shorthands(). As with symmetric functions, you do not need to run this command 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
Injecting QS as shorthand for Quasisymmetric functions over the Rational Field in the Quasisymmetric Schur 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 \(x_1, x_2, \ldots, x_m\):

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

Working with symmetric functions

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 \(m_\lambda = \sum_{\mathrm{sort}(c) = \lambda} M_c\), where \(\mathrm{sort}(c)\) means the partition obtained by sorting the composition \(c\):

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 \(f\) 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 \(f\) is symmetric, there are methods to convert \(f\) 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)

QSym is a Hopf algebra

The product on \(QSym\) 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 \(QSym\) with the structure of a Hopf algebra. It is the dual Hopf algebra of the non-commutative symmetric functions \(NCSF\). Under this duality, the Monomial basis of \(QSym\) is dual to the Complete basis of \(NCSF\), and the Fundamental basis of \(QSym\) is dual to the Ribbon basis of \(NCSF\) (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 \(H\) and \(G\) be elements of \(QSym\) and \(h\) an element of \(NCSF\). Then if we represent the duality pairing with the mathematical notation \([ \cdot, \cdot ]\), we have:

\[[H \cdot G, h] = [H \otimes G, \Delta(h)].\]

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 \(H\) is an element of \(QSym\) and \(g\) and \(h\) are elements of \(NCSF\), then

\[[ H, g \cdot h ] = [ \Delta(H), g \otimes h ].\]

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 \(g \in NCSF\) and consider the linear endomorphism of \(NCSF\) defined by left (respectively, right) multiplication by \(g\). Since there is a duality between \(QSym\) and \(NCSF\), this linear transformation induces an operator \(g^\perp\) on \(QSym\) satisfying

\[[ g^\perp(H), h ] = [ H, g \cdot h ].\]

for any non-commutative symmetric function \(h\).

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, the following identities:

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 \(0\) unless the composition \(J\) begins with \(I\) and M(J).skew_by(S(I), side='right') is \(0\) unless the composition \(J\) ends with \(I\):

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 \(I\) to \(-1\) to the size of \(I\) times the Fundamental basis element indexed by the conjugate composition to \(I\):

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 \(q = 0\). Krob and Thibon. Journal of Algebraic Combinatorics 6 (1997), 339-376.