#15150: Implement NCSym
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Reporter: tscrim | Owner: sage-combinat
Type: enhancement | Status: needs_review
Priority: major | Milestone: sage-5.12
Component: combinatorics | Resolution:
Keywords: | Merged in:
Authors: Travis Scrimshaw | Reviewers:
Report Upstream: N/A | Work issues:
Branch: | Commit:
Dependencies: #15143 | Stopgaps:
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Old description:
> Implement the Hopf algebra of symmetric functions in non-commuting
> variables in the following bases:
>
> - monomial '''m'''
> - elementary '''e'''
> - homogeneous '''h'''
> - power-sum '''p'''
> - '''x'''
> - '''q'''
>
> as well as the dual basis '''w'''.
New description:
Implement the Hopf algebra of symmetric functions in non-commuting
variables in the following bases:
- monomial '''m'''
- elementary '''e'''
- homogeneous '''h'''
- power-sum '''p'''
- '''x'''
- '''q'''
as well as the dual basis '''w'''.
Apply: [attachment:trac_15150-ncsym-ts.patch]
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Comment (by tscrim):
Here's the new version of the patch which (hopefully) takes care of all of
Darij's comments. How the `internal_coproduct*` works is a trick with
`lazy_attribute` which `*_by_coercion()` creates the correct morphism
object if `*_on_basis()` is not implemented and so it behaves like a
method.
I've removed the coercion from `NCSym` to `Sym` since it is not a Hopf
morphism, which is a requirement for it to be a coercion. I've fixed the
input to accept `x([[1,3],[2]])` as well.
Currently to get from `x` to `q` (or vice versa) there's a few coercions
going on, and I don't know of a way to go between them directly. (Data
suggests that `x(q[A])` is a sum over some subset of set partitions with
all coefficients occurring (-1)^nest(A)^.)
For `DNCSym`, is there a way to express an element in the '''w''' basis as
a polynomial? Currently I have a map from `Sym` to `DNCSym` which is at
least (appears to me to be) an algebra morphism (see the
`sum_of_partitions()` method). Is this a Hopf morphism and is the map from
`DNCSym` to `Sym` the inverse of this map?
Thank you all for looking at this patch,[[BR]]
Travis
--
Ticket URL: <http://trac.sagemath.org/ticket/15150#comment:20>
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