Opened 5 years ago

# FQSym (Malvenuto-Reutenauer) Hopf algebras

Reported by: Owned by: elixyre major sage-duplicate/invalid/wontfix combinatorics alauve, zabrocki, amypang, darij, kdilks, sage-combinat, tscrim Jean-Baptiste Priez N/A public/hopf_algebras/fqsym (Commits) 358423348671721e1987cf99e2be1237ab0bbd45 #19264

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[]
```

• #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

### comment:2 Changed 5 years ago by amypang

• Type changed from PLEASE CHANGE to enhancement

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

Amy

### comment:3 Changed 5 years ago by elixyre

• Description modified (diff)

### comment:4 Changed 3 years ago by chapoton

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

### comment:5 Changed 3 years ago by chapoton

see #23987 for a minimal implementation of FQSYM

### comment:6 Changed 3 years ago by chapoton

• Milestone changed from sage-6.9 to sage-duplicate/invalid/wontfix

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

### comment:7 Changed 3 years ago by chapoton

• Status changed from new to needs_review

### comment:8 Changed 3 years ago by gh-darijgr

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

### comment:9 Changed 3 years ago by chapoton

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

### comment:10 Changed 3 years ago by 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.

### comment:11 Changed 20 months ago by chapoton

• Status changed from needs_review to needs_work
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