FQSym (Malvenuto-Reutenauer) Hopf algebras

[elixyre]

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

[amypang]

Thank you very much for your hard work Jean-Baptiste!

Amy

[chapoton]

for future work, I have made a bare branch at "u/chapoton/19278"

[chapoton]

see #23987 for a minimal implementation of FQSYM

[chapoton]

probably we can now close this one as duplicate / invalid ?

[gh-darijgr]

The branch is red. Is there a quick way to see the "real" diff?

[chapoton]

clicking on "commits", you will see that only the last commit is pertinent to this ticket, all the others come from the dependency

[gh-darijgr]

Ah. Well, it isn't fully duplicate, since it implements several bases we don't have, and also implements dendriform *co*products. But I don't think I will have time to catch up on this in foreseeable time. And I am not sure how much of this ticket we can reuse, seeing that the code isn't very... polished.