#19278: FQSym (Malvenuto-Reutenauer) Hopf algebras
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Reporter: elixyre | Owner:
Type: enhancement | Status: new
Priority: major | Milestone: sage-6.9
Component: combinatorics | Resolution:
Keywords: | Merged in:
Authors: Jean-Baptiste | Reviewers:
Priez | Work issues:
Report Upstream: N/A | Commit:
Branch: | 358423348671721e1987cf99e2be1237ab0bbd45
public/hopf_algebras/fqsym | Stopgaps:
Dependencies: #19264 |
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Description changed by elixyre:
Old description:
> A first version of the Hopf algebra of Malvenuto-Reutenauer.
>
> This ticket provides several bases and the product and the coproduct.
> Other operators should be provided in others tickets (bidendriform, inner
> product, scalar product, polynomial realizations, etc).
>
> Those tickets should be create soon (the tickets number should be
> inventoried here).
>
> {{{
> sage: F = FQSym(QQ).F()
> sage: F[3,1,2] * F[1,2]
> F[3, 1, 2, 4, 5] + F[3, 1, 4, 2, 5] + F[3, 1, 4, 5, 2] + F[3, 4, 1, 2, 5]
> + F[3, 4, 1, 5, 2] + F[3, 4, 5, 1, 2] + F[4, 3, 1, 2, 5] + F[4, 3, 1, 5,
> 2] + F[4, 3, 5, 1, 2] + F[4, 5, 3, 1, 2]
>
> sage: F[3,1,2].coproduct()
> F[] # F[3, 1, 2] + F[1] # F[1, 2] + F[2, 1] # F[1] + F[3, 1, 2] # F[]
> }}}
New description:
A first version of the Hopf algebra of Malvenuto-Reutenauer.
This ticket provides several bases and the product and the coproduct.
Other operators should be provided in others tickets (bidendriform, inner
product, scalar product, polynomial realizations, etc).
Those tickets should be create soon (the tickets number should be
inventoried here).
{{{
sage: F = FQSym(QQ).F()
sage: F[3,1,2] * F[1,2]
F[3, 1, 2, 4, 5] + F[3, 1, 4, 2, 5] + F[3, 1, 4, 5, 2] + F[3, 4, 1, 2, 5]
+ F[3, 4, 1, 5, 2] + F[3, 4, 5, 1, 2] + F[4, 3, 1, 2, 5] + F[4, 3, 1, 5,
2] + F[4, 3, 5, 1, 2] + F[4, 5, 3, 1, 2]
sage: F[3,1,2].coproduct()
F[] # F[3, 1, 2] + F[1] # F[1, 2] + F[2, 1] # F[1] + F[3, 1, 2] # F[]
}}}
Tickets linked:
- #19287 : FQSym with scalar product
- #19286 : FQSym with polynomial realizations
- #19285 : FQSym with diese product
- (#19284 : not mine... octave vectors...)
- #19283 : FQSym with dendriform structure
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Ticket URL: <http://trac.sagemath.org/ticket/19278#comment:3>
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