#15150: Implement NCSym
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Reporter: tscrim | Owner: sage-combinat
Type: enhancement | Status: needs_review
Priority: major | Milestone: sage-5.13
Component: combinatorics | Resolution:
Keywords: | Merged in:
Authors: Travis Scrimshaw | Reviewers:
Report Upstream: N/A | Work issues:
Branch: | Commit:
Dependencies: #15143, #15164 | Stopgaps:
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Comment (by zabrocki):
`from_symmetric_function` is already not a coercion. It is just a method.
Are you saying that you don't want to make the map from Sym -> NCSym^*^ a
coercion? It preserves the Hopf structure.
For NSym/QSym I believe we have the following:
* NSym -> Sym is a method `to_symmetric_function`
* Sym -> QSym is a coercion and can be accessed by `B( sfelement )` where
`B` is a basis of QSym
* QSym -> Sym under the condition that the element in QSym `is_symmetric`
by `qsymelement.to_symmetric_function()` or (and this is not preferred)
`m( qsymelement )`
All of these maps preserve the Hopf algebra structure. Similarly for
NCSym/NCSym^*^ we have
* NCSym -> Sym is a method `to_symmetric_function`
* Sym -> NCSym^*^ is a coercion and can be accessed by `w( sfelement )`
* NCSym^*^ -> Sym under the condition that the element in NCSym^*^
`is_symmetric` by `ncsymdelement.to_symmetric_function()` or (and this is
not preferred) `h( ncsymdelement )`
The maps NCSym -> Sym and NSym -> Sym are not coercions since these maps
are projections and hence not invertible. The inclusions of Sym ->
NCSym^*^ and Sym -> QSym have an inverse (one sided) and so I think that
they can be coercions. I am not too concerned that `p( ncsymdelement )`
doesn't work. The reason the portal through the `h` basis works is
because of the connection with the '''w''' basis.
--
Ticket URL: <http://trac.sagemath.org/ticket/15150#comment:74>
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