#19594: Implement the cactus group
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Reporter: tscrim | Owner: tscrim
Type: enhancement | Status: needs_review
Priority: major | Milestone: sage-7.0
Component: group theory | Resolution:
Keywords: cactus | Merged in:
Authors: Travis Scrimshaw | Reviewers:
Report Upstream: N/A | Work issues:
Branch: | Commit:
public/groups/cactus_group-19594 | 7eb2a1278ea0ca08375f87e0f82081218a2ea1ec
Dependencies: | Stopgaps:
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Comment (by tscrim):
Replying to [comment:23 dimpase]:
> Replying to [comment:21 tscrim]:
> > `s14 |-> (1,4) (2,3)` is the correct image as `s14` should correspond
to flipping the entire interval `[1, 4]`.
>
> oops, OK then. Sorry for noise. By the way, what is proper Sage way to
get words generating
> the kernel of the homomorphism to S_n?
What I ended up doing on this ticket is just iterating over the ambient
group J,,n,, and returning the element in the kernel.
> > I asked Peter Tingley, who asked Joel Kamnitzer, and they do not know
of a normal form.
> >
> > What do both of you think about trying to prove an analog of
Matsumoto's theorem for the cactus group, i.e., that all reduced words are
related to each other by just applying the defining relations, as per
comment:15?
>
> this looks a bit unlikely that a Matsumoto's-type argument would work
here, with so many redundant generators (as opposed to the case of Coxeter
groups). What might work is a kind of argument one sees for presentations
of soluble groups, where one has commutator relations, allowing one to
"sort" generators in words; but as we have S_n acting, unsolvable for
n>4...
I am guessing it would be too much to ask for a Garside-type structure
coming from the projection onto S,,n,,, where every element can be written
as `xy^k` where `x` is the natural section of S,,n,, and `y` is some
special element in J,,n,, (similar to a normal form for the braid group
elements, where `y` corresponds to the section of the long element).
Are you thinking that there might be reduced expressions which require
using the `x^2 = 1` identity to add letters temporarily to get between
them for n > 4?
I'm also emailing Joel and Peter about possible assistance we might get
from geometric information.
--
Ticket URL: <http://trac.sagemath.org/ticket/19594#comment:24>
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