#11929: Implement quasi-symmetric functions
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Reporter: bruce | Owner: jbandlow
Type: enhancement | Status: new
Priority: minor | Milestone: sage-4.7.3
Component: combinatorics | Keywords: Hopf algebras
Work_issues: | Upstream: N/A
Reviewer: bruce | Author: bruce
Merged: | Dependencies:
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Comment(by jbandlow):
Replying to [comment:4 bruce]:
> That looks great! I have read the code but not tried to run it. I will
need to see if I can follow the instructions in the on-line documentation.
I have some comments:
It turns out you won't be able to run it, as it depends on some category
code which is also only available on the combinat queue for the time
being. I'll check with Nicolas about how to proceed on this.
> * You have declared quasisymmetric functions as Hopf algebra but I can
only see algebra operations. (This is sufficient for my purposes.)
Good catch! It might not be a bad idea for me to implement the basic Hopf
operations here as well; it shouldn't be too difficult, I think.
> * You have implemented the inclusion of symmetric functions by writing
the monomial functions in terms of the monomial functions. It is also
straightforward to implement this by writing the elementary and
homogeneous functions in terms of the fundamental functions. I don't know
what, if anything, is gained by implementing this.
Implementing the various homomorphisms could definitely improve
performance in some cases, but I'm less inclined to go down that road at
the moment.
> * Do you mean to allow the base ring to be a non-commutative ring?
Another good catch. I definitely haven't thought through the issues of
that. Perhaps I should restrict to commutative rings.
> I asked for the principal specialisation but did not give the whole
story. Let f be of degree r and ps(f) the principle specialisation. Then
what I am actually interested in is (1-q)...(1-q^r^)ps(f) which is a
polynomial in q. This is straightforward to implement using the
major_index method of Composition. I have got in the habit of calling this
the fake degree polynomial.
>
> Mathematically, this is a graded algebra homomorphism to the quantum
divided power algebra. This is a graded Z[q]-algebra. As a graded
Z[q]-module it is Z[q,x] where x has degree one. The multiplication is
x^r^.x^s^ = [r+s,r]_q x^r+s^ (where [r+s,r]_q is the quantum binomial
coefficient).
>
> I can't see a coproduct on the quantum divided power algebra which makes
it a bialgebra.
>
> I have not been able to find divided power algebras (quantum or
otherwise) in sage.
> Did I miss something? and, if not, should these be implemented?
I'm not aware of the divided power algebras in sage. You may be able to
get information or ideas on implementation from the sage-algebra list.
From this description, it doesn't look like a quick and limited
implementation would be so hard. (Also, did you mean to have an x
somewhere in your definition of the homomorphism
Qsym-->QuantumDividedPowerAlgebra ? If not, I'm confused as to how the
homomorphism is graded.)
--
Ticket URL: <http://trac.sagemath.org/sage_trac/ticket/11929#comment:5>
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