#19594: Implement the cactus group
-------------------------------------+-------------------------------------
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:
-------------------------------------+-------------------------------------
Comment (by tscrim):
Replying to [comment:26 dimpase]:
> Replying to [comment:25 tscrim]:
> > Also Remark 6.2.4 of http://arxiv.org/abs/math/0203127 says that this
should be a linear group (i.e., admits a faithful finite-dimensional
representation). I should try to understand the representation they
construct and implement that as well...
>
> I don't see an immediate connection, as the presentation there seems to
include more relations, of the form `(xy)^m=1` ?
According to R. Scott's paper
[http://www.ams.org/journals/tran/2008-360-08/S0002-9947-08-04452-8/S0002-9947-08-04452-8.pdf
Right-angled Mock reflections and mock Artin groups], the cactus group is
a special case of the presentation in that paper. I still need to verify
it because I don't quite understand the first yet, but I believe Scott's
comment.
> Anyhow, it might be quite hard to prove that an f.p. group is linear ---
you probably heard the story of braid groups in this respect...
Yea, but here I think we are closer to the Coxeter group than the braid
group (and even have a representation to test). Yet, I do agree that this
could be a hard thing to prove.
I forgot to mention it, but thank you Darij for testing it and the example
showing my proposed normal form won't work.
--
Ticket URL: <http://trac.sagemath.org/ticket/19594#comment:28>
Sage <http://www.sagemath.org>
Sage: Creating a Viable Open Source Alternative to Magma, Maple, Mathematica,
and MATLAB
--
You received this message because you are subscribed to the Google Groups
"sage-trac" group.
To unsubscribe from this group and stop receiving emails from it, send an email
to [email protected].
To post to this group, send email to [email protected].
Visit this group at https://groups.google.com/group/sage-trac.
For more options, visit https://groups.google.com/d/optout.
