#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 bruce):
Replying to [comment:1 jbandlow]:
> I've attached the patch that I've been using from the sage-combinat
queue, modified to apply cleanly to sage-4.7.1. There are obviously
documentation issues, which I will begin working on. Comments welcome!
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:
* You have declared quasisymmetric functions as Hopf algebra but I can
only see algebra operations. (This is sufficient for my purposes.)
* 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.
* Do you mean to allow the base ring to be a non-commutative ring?
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?
--
Ticket URL: <http://trac.sagemath.org/sage_trac/ticket/11929#comment:4>
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