#17792: Word problem for FareySymbol
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Reporter: mmasdeu | Owner:
Type: enhancement | Status: needs_review
Priority: major | Milestone: sage-6.7
Component: modular forms | Resolution:
Keywords: farey symbol, | Merged in:
SL2Z, word problem | Reviewers:
Authors: Marc Masdeu | Work issues:
Report Upstream: N/A | Commit:
Branch: | 18e034409e8b62c81821e93132a11dd15d7fd432
public/ticket/17792 | Stopgaps:
Dependencies: |
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Comment (by vdelecroix):
Replying to [comment:7 mmasdeu]:
> What you say makes a lot of sense, but the representation used here is
quite standard in other implementations of finitely presented groups
(AFAIK), and also in Sage. It's called Tietze representation, see
[http://www.sagemath.org/doc/reference/groups/sage/groups/finitely_presented.html].
Since unfortunately -0 = 0, the 1-based approach is essential here no
matter how much you like 0-based languages :-).
Quoting Sage in order to prove that you are right is wrong ;-) This Tietze
representation is AFAIK something internal to GAP system (which is
1-based). IMHO The fact that we used it as a non hidden attribute in free
group is wrong. I did not find any mathematical reference mentioning
"Tietze representation".
When somebody calls "word problem" he/she is even expecting a list of
generators! Using the (possibly more compact) list of `(gen, power)` is
fine. Replacing `gen` with its index is already a little bit of a hack.
Replacing `gen` with its index plus one is too much for calling it
`word_problem`. IMHO, the best would be a `Factorization` object
{{{#!python
sage: sage: F.<x,y> = FreeGroup()
sage: f = Factorization([(x,2), (y,-3), (x,1), (y,1)], sort=False)
sage: f
x^2 * y^-3 * x * y
sage: for i,j in f:
....: print i,j
x 2
y -3
x 1
y 1
}}}
For matrices, the representation is ugly, but this can be easily fixed to
be something like
{{{#!python
sage: m1 = SL2Z([1,1,0,1])
sage: m2 = SL2Z([1,0,1,1])
sage: f = Factorization([(m1,2),(m2,-3),(m1,2)], sort=False)
sage: f # this is currently ugly
[1 1]^2 * [1 0]^-3 * [1 1]^2
[0 1] [1 1] [0 1]
sage: prod(f) # this does work
[-5 -8]
[-3 -5]
}}}
I would be happy to fix the `Factorization._repr_` if you like this
solution.
> Also, you can always do something like:
> {{{
> sage: wd = F.word_problem(g)
> sage: prod(F.gen(i-1) if i > 0 else F.gen(-i-1)**-1 for i in wd) == g
> True
> }}}
Very elegant and very natural!
Vincent
--
Ticket URL: <http://trac.sagemath.org/ticket/17792#comment:8>
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