Opened 3 years ago

# Fix few bugs in ClusterAlgebra discovered while working on #26771 — at Version 3

Reported by: Owned by: etn40ff major sage-8.9 algebra ClusterAlgebras tscrim, gmoose05, stumpc5, gh-EBanaian, drupel, gh-kelleye Salvatore Stella N/A u/etn40ff/28176 d3cd59d86bf16c8414d36cdf5d95d2c5abd7d07f

As pointed out by the people at #26771, `ClusterAlgebra` has few bugs.

• Default arguments do not play nicely with `UniqueRepresentation`

```  sage: A1 = ClusterAlgebra(['A',2])
sage: A2 = ClusterAlgebra(['A',2], cluster_variable_prefix='x')
sage: A1 is A2
False
```
• `mutate_initial` keeps the same variable names; In theory this is not a problem but may create confusion. For example suppose we are looking at the Grassmannian of planes in a 4 dimensional vector space. Then the associate cluster algebra is
```  sage: A = ClusterAlgebra( matrix(5,[0,1,-1,1,-1]), cluster_variable_names=['p13'], coefficient_names=['p12','p23','p34','p41'], scalars=QQbar); A
A Cluster Algebra with cluster variable p13 and coefficients p12, p23, p34, p41 over Algebraic Field
```

Swapping out `p13` from the initial seed should give `p24` or, at the very least *not* `p13`. Getting `p24` requires some understanding of what the ring in question really is that goes beyond the scope of this class. Currently we get

```  sage: A.mutate_initial(0)
A Cluster Algebra with cluster variable p13 and coefficients p12, p23, p34, p41 over Algebraic Field
```
• `mutate_initial` does not compute all the required F-polynomials and it does not enforce the fact that they are not rational expressions
```  sage: A1 = ClusterAlgebra(['A',[2,1],1])
sage: A2 = A1.mutate_initial([0,1,0])
sage: len(A2.g_vectors_so_far()) == len(A2.F_polynomials_so_far())
False
sage: all(parent(f) == A2._U for f in A2.F_polynomials_so_far())
False
```
• finally `mutate_initial` does not enforce the fact that initial g-vectors belong to the initial cluster but rather prepend the performed mutation sequence to all the paths already known.
```  sage: A2.find_g_vector((0,0,1))
[0, 1, 0]
```

This, while technically not false, may generate confusion.

• `_mutated_F` is doing `/` rather than `//` resulting in slower computations and parent issues.

This ticket deals with all these issues.

### comment:1 Changed 3 years ago by etn40ff

• Branch set to u/etn40ff/28176
• Cc tscrim gmoose05 stumpc5 gh-EBanaian drupel gh-kelleye added
• Commit set to d3cd59d86bf16c8414d36cdf5d95d2c5abd7d07f
• Component changed from PLEASE CHANGE to algebra
• Status changed from new to needs_review

New commits:

 ​d3cd59d `Fix 28176`

### comment:2 Changed 3 years ago by chapoton

bot is not happy:

```sage -t --long src/sage/algebras/cluster_algebra.py  # Tab character found
```

### comment:3 Changed 3 years ago by etn40ff

• Description modified (diff)

Oops. Fixed, sorry Frédéric!

With the occasion I also got some dramatic speedup in `mutate_initial` by not mutating F-polynomials a quadratic number of times.

Also I changed a division into a polynomial division in `_mutated_F`; this enforces F-polynomials to live in the correct ambient and yields a decent speedup in `mutate`.

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