In this discussion, we came up with a rough draft of an interface for the method returning the dual of an object. Here is a summary by way of docstrings for the methods:
def dual(self, category=None):
r"""
The dual of ``self``.
By default, the dual is computed in the category
``self.category()``. If the user specifies a category, the dual will
be computed in that category.
INPUT:
- ``category`` -- category (default: the category of ``self``).
OUTPUT:
- The dual of ``self``.
EXAMPLES:
The Hopf algebra of symmetric functions is a self-dual Hopf
algebra::
sage: Sym = SymmetricFunctions(QQ); Sym
Symmetric Functions over Rational Field
sage: Sym.dual()
Symmetric Functions over Rational Field
sage: Sym.dual() is Sym
True
If we view ``Sym`` as an algebra, then its dual is a co-algebra::
sage: C = Sym.dual(category=Algebras(QQ)).category()
Category of duals of algebras over Rational Field
sage: C.super_categories()
[Category of coalgebras over Rational Field,
Category of duals of vector spaces over Rational Field]
The Schur basis for symmetric functions is self-dual and the
homogeneous symmetric functions are dual to the monomial
symmetric functions::
sage: s = Sym.schur()
sage: s.dual() is s
True
sage: h = Sym.homogeneous()
sage: m = Sym.monomial()
sage: h.dual() is m
True
Note that in the above, ``s`` (as well as ``h`` and ``m``) are Hopf
algebras with basis. Hence, their duals are also Hopf algebras with
basis.
The Hopf algebra of quasi-symmetric functions is dual, as a Hopf
algebra, to the Hopf algebra of non-commutative symmetric
functions::
sage: NCSF = NonCommutativeSymmetricFunctions(QQ)
sage: NCSF.dual()
Quasisymmetric functions over the Rational Field
::
sage: QSym = QuasiSymmetricFunctions(QQ)
sage: QSym.dual()
Non-Commutative Symmetric Functions over the Rational Field
"""
return NotImplemented
def duality_pairing(self, x, y):
r"""
The duality pairing between elements of NSym and elements of QSym.
INPUT:
- ``x`` -- an element of ``self``
- ``y`` -- an element in the dual basis of ``self``
OUTPUT:
- The result of pairing the element ``x`` of ``self`` with the
element ``y`` of the dual of ``self``.
EXAMPLES:
The Schur basis of symmetric functions is self-dual::
sage: Sym = SymmetricFunctions(QQ)
sage: s = Sym(QQ).schur()
sage: s.dual() is s
True
sage: s.duality_pairing(s[2,1,1], s[2,1,1])
1
sage: s.duality_pairing(s[2,1], s[3])
0
The fundamental basis of quasi-symmetric functions is dual to the
ribbon basis of non-commutative symmetric functions::
sage: R = NonCommutativeSymmetricFunctions(QQ).Ribbon()
sage: F = QuasiSymmetricFunctions(QQ).Fundamental()
sage: R.duality_pairing(R[1,1,2], F[1,1,2])
1
sage: R.duality_pairing(R[1,2,1], F[1,1,2])
0
sage: F.duality_pairing(F[1,2,1], R[1,1,2])
0
"""
return NotImplemented
A rudimentary implementation for duality_pairing
can be found at #8899, but see also the scalar product code for symmetric functions.
I think a bunch of the code for duality for symmetric functions can be refactored. See sage.combinat.sf.dual.
Tickets:
- #34621 Method
dual_pairing
for modules in sage.tensor
Simon raised the following question in the thread: