Sage: Ticket #19594: Implement the cactus group
https://trac.sagemath.org/ticket/19594
<p>
We implement the cactus group <em>J</em><sub>n</sub> (of type A). The cactus group has important applications in category and representation theory.
</p>
<p>
This group is not available in GAP as far as I can tell.
</p>
en-usSagehttps://trac.sagemath.org/chrome/site/logo_sagemath_trac.png
https://trac.sagemath.org/ticket/19594
Trac 1.1.6tscrimTue, 17 Nov 2015 23:18:16 GMTstatus changed; commit, branch set
https://trac.sagemath.org/ticket/19594#comment:1
https://trac.sagemath.org/ticket/19594#comment:1
<ul>
<li><strong>status</strong>
changed from <em>new</em> to <em>needs_review</em>
</li>
<li><strong>commit</strong>
set to <em>83a2cc64551145946032112169f011ef897716c3</em>
</li>
<li><strong>branch</strong>
set to <em>public/groups/cactus_group-19594</em>
</li>
</ul>
<p>
New commits:
</p>
<table class="wiki">
<tr><td><a class="ext-link" href="http://git.sagemath.org/sage.git/commit/?id=83a2cc64551145946032112169f011ef897716c3"><span class="icon"></span>83a2cc6</a></td><td><code>Implementation of the (type A) cactus group.</code>
</td></tr></table>
TicketdimpaseWed, 23 Dec 2015 20:35:21 GMT
https://trac.sagemath.org/ticket/19594#comment:2
https://trac.sagemath.org/ticket/19594#comment:2
<p>
Can you also construct the homomorphism to the symmetric group, and its kernel (i.e. pure cactus group) ?
</p>
TickettscrimThu, 24 Dec 2015 23:10:15 GMTstatus, milestone changed; cc set
https://trac.sagemath.org/ticket/19594#comment:3
https://trac.sagemath.org/ticket/19594#comment:3
<ul>
<li><strong>status</strong>
changed from <em>needs_review</em> to <em>needs_work</em>
</li>
<li><strong>cc</strong>
<em>ptingley</em> added
</li>
<li><strong>milestone</strong>
changed from <em>sage-6.10</em> to <em>sage-7.0</em>
</li>
</ul>
<p>
Replying to <a class="ticket" href="https://trac.sagemath.org/ticket/19594#comment:2" title="Comment 2">dimpase</a>:
</p>
<blockquote class="citation">
<p>
Can you also construct the homomorphism to the symmetric group, and its kernel (i.e. pure cactus group) ?
</p>
</blockquote>
<p>
I can definitely add a default coercion to the symmetric group. I should be able to do a general subgroup defined by a particular condition (such as the kernel of a morphism). I will do this now.
</p>
TicketgitFri, 25 Dec 2015 02:43:27 GMTcommit changed
https://trac.sagemath.org/ticket/19594#comment:4
https://trac.sagemath.org/ticket/19594#comment:4
<ul>
<li><strong>commit</strong>
changed from <em>83a2cc64551145946032112169f011ef897716c3</em> to <em>832d7f0de192e519ed8fd251eb8039722fad48b6</em>
</li>
</ul>
<p>
Branch pushed to git repo; I updated commit sha1. New commits:
</p>
<table class="wiki">
<tr><td><a class="ext-link" href="http://git.sagemath.org/sage.git/commit/?id=55d0e2831163c18049d5c255de2236f218e7ac69"><span class="icon"></span>55d0e28</a></td><td><code>Merge branch 'public/groups/cactus_group-19594' of trac.sagemath.org:sage into public/groups/cactus_group-19594</code>
</td></tr><tr><td><a class="ext-link" href="http://git.sagemath.org/sage.git/commit/?id=83220d668340ed074946cceeec305d6ef6139ae0"><span class="icon"></span>83220d6</a></td><td><code>Implement coercion/maps from the cactus group to the permutation group.</code>
</td></tr><tr><td><a class="ext-link" href="http://git.sagemath.org/sage.git/commit/?id=0b3b875cc0411103b0f03df44a1519be290aa35d"><span class="icon"></span>0b3b875</a></td><td><code>Started implemention of kernel subgroups.</code>
</td></tr><tr><td><a class="ext-link" href="http://git.sagemath.org/sage.git/commit/?id=832d7f0de192e519ed8fd251eb8039722fad48b6"><span class="icon"></span>832d7f0</a></td><td><code>Added kernel subgroup and pure cactus group.</code>
</td></tr></table>
TickettscrimFri, 25 Dec 2015 02:44:09 GMTstatus changed
https://trac.sagemath.org/ticket/19594#comment:5
https://trac.sagemath.org/ticket/19594#comment:5
<ul>
<li><strong>status</strong>
changed from <em>needs_work</em> to <em>needs_review</em>
</li>
</ul>
<p>
Done and done.
</p>
TicketgitMon, 28 Dec 2015 02:01:44 GMTcommit changed
https://trac.sagemath.org/ticket/19594#comment:6
https://trac.sagemath.org/ticket/19594#comment:6
<ul>
<li><strong>commit</strong>
changed from <em>832d7f0de192e519ed8fd251eb8039722fad48b6</em> to <em>7eb2a1278ea0ca08375f87e0f82081218a2ea1ec</em>
</li>
</ul>
<p>
Branch pushed to git repo; I updated commit sha1. New commits:
</p>
<table class="wiki">
<tr><td><a class="ext-link" href="http://git.sagemath.org/sage.git/commit/?id=7c0fdf336a1948b67366330c6469d2f777334033"><span class="icon"></span>7c0fdf3</a></td><td><code>Merge branch 'public/groups/cactus_group-19594' of git://trac.sagemath.org/sage into cactus</code>
</td></tr><tr><td><a class="ext-link" href="http://git.sagemath.org/sage.git/commit/?id=7eb2a1278ea0ca08375f87e0f82081218a2ea1ec"><span class="icon"></span>7eb2a12</a></td><td><code>correct quadratic relations</code>
</td></tr></table>
TicketdarijMon, 28 Dec 2015 02:02:44 GMT
https://trac.sagemath.org/ticket/19594#comment:7
https://trac.sagemath.org/ticket/19594#comment:7
<p>
Travis -- I like the fact that you are implementing this group, but are you sure that the switches you do in <code>_product_on_gens</code> are enough to bring every word in its generators to a normal form? I.e., that any two products of generators that compare as un-equal are actually distinct?
</p>
TickettscrimMon, 28 Dec 2015 04:56:08 GMT
https://trac.sagemath.org/ticket/19594#comment:8
https://trac.sagemath.org/ticket/19594#comment:8
<p>
Replying to <a class="ticket" href="https://trac.sagemath.org/ticket/19594#comment:7" title="Comment 7">darij</a>:
</p>
<blockquote class="citation">
<p>
Travis -- I like the fact that you are implementing this group, but are you sure that the switches you do in <code>_product_on_gens</code> are enough to bring every word in its generators to a normal form? I.e., that any two products of generators that compare as un-equal are actually distinct?
</p>
</blockquote>
<p>
I haven't formally proved it (nor do I know if there is a formal proof that there is a normal form). I'm inclined to believe it works, but I'm not 100% sure (nor can I prove it right now). Perhaps we should add a warning about this?
</p>
TicketdimpaseMon, 28 Dec 2015 07:02:04 GMT
https://trac.sagemath.org/ticket/19594#comment:9
https://trac.sagemath.org/ticket/19594#comment:9
<p>
Is any cactus group automatic? GAP has <a class="ext-link" href="http://www.gap-system.org/Packages/kbmag.html"><span class="icon"></span>means</a> to try to check whether an f.p. group is automatic. (this would give one a normal form, etc).
</p>
TickettscrimMon, 28 Dec 2015 08:11:26 GMT
https://trac.sagemath.org/ticket/19594#comment:10
https://trac.sagemath.org/ticket/19594#comment:10
<p>
Replying to <a class="ticket" href="https://trac.sagemath.org/ticket/19594#comment:9" title="Comment 9">dimpase</a>:
</p>
<blockquote class="citation">
<p>
Is any cactus group automatic? GAP has <a class="ext-link" href="http://www.gap-system.org/Packages/kbmag.html"><span class="icon"></span>means</a> to try to check whether an f.p. group is automatic. (this would give one a normal form, etc).
</p>
</blockquote>
<p>
I don't know. The best I could find from some quick Googling is <a class="ext-link" href="http://mathoverflow.net/questions/65280/splitting-of-homomorphism-from-cactus-group-to-permutation-group"><span class="icon"></span>this MO question</a> (which suggests that my approach does not give a normal form). We can try as I also included a f.p. version of the group which should be easy to feed to GAP. However, from me it would have to wait until tomorrow as I'm going to sleep now.
</p>
TicketdimpaseMon, 28 Dec 2015 12:20:34 GMT
https://trac.sagemath.org/ticket/19594#comment:11
https://trac.sagemath.org/ticket/19594#comment:11
<p>
for n=3,4 Knuth-Bendix procedure (from the GAP package kbmag I mentioned in comment 9) quickly returns a rewriting system for J_n, but it gets stuck for n=5. I haven't tried to push this further yet.
</p>
TicketdimpaseMon, 28 Dec 2015 14:59:50 GMT
https://trac.sagemath.org/ticket/19594#comment:12
https://trac.sagemath.org/ticket/19594#comment:12
<p>
I wonder why the code does not compute a presentation for the pure cactus group. This is more or less standard thing, Reidemeister-Schreier algorithm, implemented in GAP (see e.g. <code>PresentationSubgroup</code>).
</p>
TicketdarijMon, 28 Dec 2015 17:18:38 GMT
https://trac.sagemath.org/ticket/19594#comment:13
https://trac.sagemath.org/ticket/19594#comment:13
<p>
After a few experiments (on paper), I have started suspecting that Travis's code *does* bring every word to a normal form. This could be a cool combinatorial result, if true.
</p>
<p>
If true, it should be provable using the diamond lemma... Does anyone volunteer to bash the cases?
</p>
TickettscrimMon, 28 Dec 2015 18:00:44 GMT
https://trac.sagemath.org/ticket/19594#comment:14
https://trac.sagemath.org/ticket/19594#comment:14
<p>
Replying to <a class="ticket" href="https://trac.sagemath.org/ticket/19594#comment:12" title="Comment 12">dimpase</a>:
</p>
<blockquote class="citation">
<p>
I wonder why the code does not compute a presentation for the pure cactus group. This is more or less standard thing, Reidemeister-Schreier algorithm, implemented in GAP (see e.g. <code>PresentationSubgroup</code>).
</p>
</blockquote>
<p>
Is this a comment on my code or GAP's? I don't know the Reidemeister-Schreier algorithm, but I am not opposed to trying to extend the implementation.
</p>
TickettscrimMon, 28 Dec 2015 18:04:36 GMT
https://trac.sagemath.org/ticket/19594#comment:15
https://trac.sagemath.org/ticket/19594#comment:15
<p>
Replying to <a class="ticket" href="https://trac.sagemath.org/ticket/19594#comment:13" title="Comment 13">darij</a>:
</p>
<blockquote class="citation">
<p>
After a few experiments (on paper), I have started suspecting that Travis's code *does* bring every word to a normal form. This could be a cool combinatorial result, if true.
</p>
<p>
If true, it should be provable using the diamond lemma... Does anyone volunteer to bash the cases?
</p>
</blockquote>
<p>
I am pretty sure that the terminating condition is possible, but I'm worried about a case of something like <code>s[4, 5] * s[1, 7] * s[5, 8]</code> and the shuffles. However, I agree it would be a nice little result if this was true, and as far as I can find, there is no such analogous result. At the very least, I think we can easily show an analog of Matsumoto's lemma, and so we build out a reduced word graph and find a lex min element in that.
</p>
TicketdarijMon, 28 Dec 2015 18:15:05 GMT
https://trac.sagemath.org/ticket/19594#comment:16
https://trac.sagemath.org/ticket/19594#comment:16
<p>
Sorry, Travis, you are right with your worries:
</p>
<pre class="wiki">sage: s45*s13 == s13*s45
True
sage: s26*s45*s13 == s26*s13*s45
False
</pre><p>
This cactus is thornier than I thought.
</p>
TicketdimpaseMon, 28 Dec 2015 20:38:44 GMT
https://trac.sagemath.org/ticket/19594#comment:17
https://trac.sagemath.org/ticket/19594#comment:17
<p>
Replying to <a class="ticket" href="https://trac.sagemath.org/ticket/19594#comment:16" title="Comment 16">darij</a>:
</p>
<blockquote class="citation">
<p>
Sorry, Travis, you are right with your worries:
</p>
<pre class="wiki">sage: s45*s13 == s13*s45
True
sage: s26*s45*s13 == s26*s13*s45
False
</pre><p>
This cactus is thornier than I thought.
</p>
</blockquote>
<p>
this is also what Knuth-Bendix computation was predicting, that for n>4 things get hard.
</p>
TicketdimpaseMon, 28 Dec 2015 20:39:27 GMT
https://trac.sagemath.org/ticket/19594#comment:18
https://trac.sagemath.org/ticket/19594#comment:18
<p>
Replying to <a class="ticket" href="https://trac.sagemath.org/ticket/19594#comment:14" title="Comment 14">tscrim</a>:
</p>
<blockquote class="citation">
<p>
Replying to <a class="ticket" href="https://trac.sagemath.org/ticket/19594#comment:12" title="Comment 12">dimpase</a>:
</p>
<blockquote class="citation">
<p>
I wonder why the code does not compute a presentation for the pure cactus group. This is more or less standard thing, Reidemeister-Schreier algorithm, implemented in GAP (see e.g. <code>PresentationSubgroup</code>).
</p>
</blockquote>
<p>
Is this a comment on my code or GAP's? I don't know the Reidemeister-Schreier algorithm, but I am not opposed to trying to extend the implementation.
</p>
</blockquote>
<p>
it's a comment saying that this is doable (not sure how interesting).
</p>
TicketdimpaseMon, 28 Dec 2015 21:51:43 GMT
https://trac.sagemath.org/ticket/19594#comment:19
https://trac.sagemath.org/ticket/19594#comment:19
<p>
There seems to be something wrong with the relations you provide.I am trying to check that you indeed have a homomorphism from J_4 to Sym(4), but GAP returns <code>fail</code>. Does your code check that your map is a group homomorphism?
</p>
<pre class="wiki">gap> F:=FreeGroup(6);
<free group on the generators [ f1, f2, f3, f4, f5, f6 ]>
gap> s12:=F.1;; s13:=F.2;; s14:=F.3;; s23:=F.4;; s24:=F.5;; s34:=F.6;;
gap> rels:=[s12^2, s13^2, s14^2, s23^2, s24^2, s34^2, s13*s12*s13^-1*s23^-1, s13*s23*s13^-1*s12^-1, s14*s12*s14^-1*s34^-1, s14*s13*s14^-1*s24^-1, s14*s23*s14^-1*s23^-1, s14*s24*s14^-1*s13^-1, s14*s34*s14^-1*s12^-1, s24*s23*s24^-1*s34^-1, s24*s34*s24^-1*s23^-1, s34*s12*s34^-1*s12^-1 ];
[ f1^2, f2^2, f3^2, f4^2, f5^2, f6^2, f2*f1*f2^-1*f4^-1, f2*f4*f2^-1*f1^-1, f3*f1*f3^-1*f6^-1,
f3*f2*f3^-1*f5^-1, f3*f4*f3^-1*f4^-1, f3*f5*f3^-1*f2^-1, f3*f6*f3^-1*f1^-1, f5*f4*f5^-1*f6^-1,
f5*f6*f5^-1*f4^-1, f6*f1*f6^-1*f1^-1 ]
gap> G:=F/rels;
<fp group on the generators [ f1, f2, f3, f4, f5, f6 ]>
gap> s4:=Group([(1,2),(1,3),(1,4),(2,3),(2,4),(3,4)]);
Group([ (1,2), (1,3), (1,4), (2,3), (2,4), (3,4) ])
gap> GeneratorsOfGroup(s4);
[ (1,2), (1,3), (1,4), (2,3), (2,4), (3,4) ]
gap> f:=GroupHomomorphismByImages(G,s4);
fail
</pre>
TicketdimpaseMon, 28 Dec 2015 22:00:52 GMT
https://trac.sagemath.org/ticket/19594#comment:20
https://trac.sagemath.org/ticket/19594#comment:20
<p>
here is a direct check of the relations:
</p>
<pre class="wiki">gap> s12:=(1,2); s13:=(1,3); s14:=(1,4); s23:=(2,3); s24:=(2,4); s34:=(3,4);
(1,2)
(1,3)
(1,4)
(2,3)
(2,4)
(3,4)
gap> [s12^2, s13^2, s14^2, s23^2, s24^2, s34^2, s13*s12*s13^-1*s23^-1, s13*s23*s13^-1*s12^-1, s14*s12*s14^-1*s34^-1, s14*s13*s14^-1*s24^-1, s14*s23*s14^-1*s23^-1, s14*s24*s14^-1*s13^-1, s14*s34*s14^-1*s12^-1, s24*s23*s24^-1*s34^-1, s24*s34*s24^-1*s23^-1, s34*s12*s34^-1*s12^-1 ];
[ (), (), (), (), (), (), (), (), (2,3,4), (2,4,3), (), (1,2,3), (1,3,2), (), (), () ]
gap>
</pre><p>
e.g. <code>s14*s12*s14^-1*s34^-1</code> evaluates to (2,3,4) in Sym(4)...
</p>
TickettscrimMon, 28 Dec 2015 22:40:32 GMT
https://trac.sagemath.org/ticket/19594#comment:21
https://trac.sagemath.org/ticket/19594#comment:21
<p>
<code>s14 |-> (1,4) (2,3)</code> is the correct image as <code>s14</code> should correspond to flipping the entire interval <code>[1, 4]</code>.
</p>
<p>
I asked Peter Tingley, who asked Joel Kamnitzer, and they do not know of a normal form.
</p>
<p>
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 <a class="ticket" href="https://trac.sagemath.org/ticket/19594#comment:15" title="Comment 15">comment:15</a>?
</p>
TicketdarijMon, 28 Dec 2015 23:04:44 GMT
https://trac.sagemath.org/ticket/19594#comment:22
https://trac.sagemath.org/ticket/19594#comment:22
<p>
It sounds like the next best thing, after we know that the obvious rewrite system does not work... But I don't see how to do it.
</p>
TicketdimpaseTue, 29 Dec 2015 15:30:23 GMT
https://trac.sagemath.org/ticket/19594#comment:23
https://trac.sagemath.org/ticket/19594#comment:23
<p>
Replying to <a class="ticket" href="https://trac.sagemath.org/ticket/19594#comment:21" title="Comment 21">tscrim</a>:
</p>
<blockquote class="citation">
<p>
<code>s14 |-> (1,4) (2,3)</code> is the correct image as <code>s14</code> should correspond to flipping the entire interval <code>[1, 4]</code>.
</p>
</blockquote>
<p>
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?
</p>
<blockquote class="citation">
<p>
I asked Peter Tingley, who asked Joel Kamnitzer, and they do not know of a normal form.
</p>
<p>
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 <a class="ticket" href="https://trac.sagemath.org/ticket/19594#comment:15" title="Comment 15">comment:15</a>?
</p>
</blockquote>
<p>
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...
</p>
TickettscrimWed, 30 Dec 2015 01:13:41 GMT
https://trac.sagemath.org/ticket/19594#comment:24
https://trac.sagemath.org/ticket/19594#comment:24
<p>
Replying to <a class="ticket" href="https://trac.sagemath.org/ticket/19594#comment:23" title="Comment 23">dimpase</a>:
</p>
<blockquote class="citation">
<p>
Replying to <a class="ticket" href="https://trac.sagemath.org/ticket/19594#comment:21" title="Comment 21">tscrim</a>:
</p>
<blockquote class="citation">
<p>
<code>s14 |-> (1,4) (2,3)</code> is the correct image as <code>s14</code> should correspond to flipping the entire interval <code>[1, 4]</code>.
</p>
</blockquote>
<p>
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?
</p>
</blockquote>
<p>
What I ended up doing on this ticket is just iterating over the ambient group J<sub>n</sub> and returning the element in the kernel.
</p>
<blockquote class="citation">
<blockquote class="citation">
<p>
I asked Peter Tingley, who asked Joel Kamnitzer, and they do not know of a normal form.
</p>
<p>
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 <a class="ticket" href="https://trac.sagemath.org/ticket/19594#comment:15" title="Comment 15">comment:15</a>?
</p>
</blockquote>
<p>
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...
</p>
</blockquote>
<p>
I am guessing it would be too much to ask for a Garside-type structure coming from the projection onto S<sub>n</sub>, where every element can be written as <code>xy^k</code> where <code>x</code> is the natural section of S<sub>n</sub> and <code>y</code> is some special element in J<sub>n</sub> (similar to a normal form for the braid group elements, where <code>y</code> corresponds to the section of the long element).
</p>
<p>
Are you thinking that there might be reduced expressions which require using the <code>x^2 = 1</code> identity to add letters temporarily to get between them for n > 4?
</p>
<p>
I'm also emailing Joel and Peter about possible assistance we might get from geometric information.
</p>
TickettscrimWed, 30 Dec 2015 01:30:54 GMT
https://trac.sagemath.org/ticket/19594#comment:25
https://trac.sagemath.org/ticket/19594#comment:25
<p>
Also Remark 6.2.4 of <a class="ext-link" href="http://arxiv.org/abs/math/0203127"><span class="icon"></span>http://arxiv.org/abs/math/0203127</a> 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...
</p>
TicketdimpaseWed, 30 Dec 2015 10:51:53 GMT
https://trac.sagemath.org/ticket/19594#comment:26
https://trac.sagemath.org/ticket/19594#comment:26
<p>
Replying to <a class="ticket" href="https://trac.sagemath.org/ticket/19594#comment:25" title="Comment 25">tscrim</a>:
</p>
<blockquote class="citation">
<p>
Also Remark 6.2.4 of <a class="ext-link" href="http://arxiv.org/abs/math/0203127"><span class="icon"></span>http://arxiv.org/abs/math/0203127</a> 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...
</p>
</blockquote>
<p>
I don't see an immediate connection, as the presentation there seems to include more relations, of the form <code>(xy)^m=1</code> ?
</p>
<p>
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...
</p>
TickettscrimWed, 30 Dec 2015 15:39:46 GMT
https://trac.sagemath.org/ticket/19594#comment:27
https://trac.sagemath.org/ticket/19594#comment:27
<p>
Replying to <a class="ticket" href="https://trac.sagemath.org/ticket/19594#comment:24" title="Comment 24">tscrim</a>:
</p>
<blockquote class="citation">
<p>
I am guessing it would be too much to ask for a Garside-type structure coming from the projection onto S<sub>n</sub>, where every element can be written as <code>xy^k</code> where <code>x</code> is the natural section of S<sub>n</sub> and <code>y</code> is some special element in J<sub>n</sub> (similar to a normal form for the braid group elements, where <code>y</code> corresponds to the section of the long element).
</p>
</blockquote>
<p>
A Garside-type structure would not work here because that is used for passing from the braid monoid to the braid group. So I really don't think this will work for the cactus group.
</p>
TickettscrimWed, 30 Dec 2015 15:44:05 GMT
https://trac.sagemath.org/ticket/19594#comment:28
https://trac.sagemath.org/ticket/19594#comment:28
<p>
Replying to <a class="ticket" href="https://trac.sagemath.org/ticket/19594#comment:26" title="Comment 26">dimpase</a>:
</p>
<blockquote class="citation">
<p>
Replying to <a class="ticket" href="https://trac.sagemath.org/ticket/19594#comment:25" title="Comment 25">tscrim</a>:
</p>
<blockquote class="citation">
<p>
Also Remark 6.2.4 of <a class="ext-link" href="http://arxiv.org/abs/math/0203127"><span class="icon"></span>http://arxiv.org/abs/math/0203127</a> 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...
</p>
</blockquote>
<p>
I don't see an immediate connection, as the presentation there seems to include more relations, of the form <code>(xy)^m=1</code> ?
</p>
</blockquote>
<p>
According to R. Scott's paper <a class="ext-link" href="http://www.ams.org/journals/tran/2008-360-08/S0002-9947-08-04452-8/S0002-9947-08-04452-8.pdf"><span class="icon"></span>Right-angled Mock reflections and mock Artin groups</a>, 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.
</p>
<blockquote class="citation">
<p>
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...
</p>
</blockquote>
<p>
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.
</p>
<p>
I forgot to mention it, but thank you Darij for testing it and the example showing my proposed normal form won't work.
</p>
TicketgitSun, 03 Jan 2016 03:45:42 GMTcommit changed
https://trac.sagemath.org/ticket/19594#comment:29
https://trac.sagemath.org/ticket/19594#comment:29
<ul>
<li><strong>commit</strong>
changed from <em>7eb2a1278ea0ca08375f87e0f82081218a2ea1ec</em> to <em>a3ab2d2076464e3451720832ef384186abcc1552</em>
</li>
</ul>
<p>
Branch pushed to git repo; I updated commit sha1. New commits:
</p>
<table class="wiki">
<tr><td><a class="ext-link" href="http://git.sagemath.org/sage.git/commit/?id=a3ab2d2076464e3451720832ef384186abcc1552"><span class="icon"></span>a3ab2d2</a></td><td><code>Added DJS representation for the cactus group.</code>
</td></tr></table>
TickettscrimSun, 03 Jan 2016 03:52:51 GMT
https://trac.sagemath.org/ticket/19594#comment:30
https://trac.sagemath.org/ticket/19594#comment:30
<p>
Okay, so the cactus group corresponds to type A<sub>n</sub> and when <code>R</code> is the full power set of the index set. I've added the representation described in the DJS paper here. One related question that comes to mind is what happens when <code>t=1</code>, do we get a (faithful) representation of the right-angled Coxeter group which has the <code>t=1</code> bilinear form? We have a lot to think on for this...
</p>
<p>
The end result of our discussion as I see it is that we don't have a good way to construct normal forms of elements at present. So for the purposes of this ticket, should we include this as-is with a warning stating that elements may be equal even if <code>==</code> does not necessarily return <code>True</code>?
</p>
TicketdimpaseTue, 05 Jan 2016 11:39:14 GMT
https://trac.sagemath.org/ticket/19594#comment:31
https://trac.sagemath.org/ticket/19594#comment:31
<p>
I am not sure that it is a good idea to have this group in the same catalog as the others (which do have a normal form for its elements). As it stands, the group operation for J<sub>n</sub> is not well-defined, you know.
</p>
TicketgitThu, 26 May 2016 10:01:21 GMTcommit changed
https://trac.sagemath.org/ticket/19594#comment:32
https://trac.sagemath.org/ticket/19594#comment:32
<ul>
<li><strong>commit</strong>
changed from <em>a3ab2d2076464e3451720832ef384186abcc1552</em> to <em>3e99cfe0c0f0054ce44fa8e1ad9f2ea6c929ea2a</em>
</li>
</ul>
<p>
Branch pushed to git repo; I updated commit sha1. New commits:
</p>
<table class="wiki">
<tr><td><a class="ext-link" href="http://git.sagemath.org/sage.git/commit/?id=3e99cfe0c0f0054ce44fa8e1ad9f2ea6c929ea2a"><span class="icon"></span>3e99cfe</a></td><td><code>Merge branch 'public/groups/cactus_group-19594' into 7.3.b1</code>
</td></tr></table>
TicketgitFri, 11 Nov 2016 21:09:48 GMTcommit changed
https://trac.sagemath.org/ticket/19594#comment:33
https://trac.sagemath.org/ticket/19594#comment:33
<ul>
<li><strong>commit</strong>
changed from <em>3e99cfe0c0f0054ce44fa8e1ad9f2ea6c929ea2a</em> to <em>d883593277148ad5301c7f428efe5edf6d1585a7</em>
</li>
</ul>
<p>
Branch pushed to git repo; I updated commit sha1. New commits:
</p>
<table class="wiki">
<tr><td><a class="ext-link" href="https://git.sagemath.org/sage.git/commit/?id=d883593277148ad5301c7f428efe5edf6d1585a7"><span class="icon"></span>d883593</a></td><td><code>Merge branch 'public/groups/cactus_group-19594' in 7.5.b2</code>
</td></tr></table>
TicketgitThu, 05 Jan 2017 20:46:35 GMTcommit changed
https://trac.sagemath.org/ticket/19594#comment:34
https://trac.sagemath.org/ticket/19594#comment:34
<ul>
<li><strong>commit</strong>
changed from <em>d883593277148ad5301c7f428efe5edf6d1585a7</em> to <em>002edc4c3ba10274b93dbf960ed5a4d6d8e7f270</em>
</li>
</ul>
<p>
Branch pushed to git repo; I updated commit sha1. New commits:
</p>
<table class="wiki">
<tr><td><a class="ext-link" href="https://git.sagemath.org/sage.git/commit/?id=854f529950e18c59723c855e951b5b6e134a062d"><span class="icon"></span>854f529</a></td><td><code>Merge branch 'public/groups/cactus_group-19594' in 7.5.rc1</code>
</td></tr><tr><td><a class="ext-link" href="https://git.sagemath.org/sage.git/commit/?id=002edc4c3ba10274b93dbf960ed5a4d6d8e7f270"><span class="icon"></span>002edc4</a></td><td><code>trac 19594 pep8 cleanup, deprecation handled, py3-compatible code</code>
</td></tr></table>
TicketBruceFri, 14 Jul 2017 14:46:56 GMT
https://trac.sagemath.org/ticket/19594#comment:35
https://trac.sagemath.org/ticket/19594#comment:35
<p>
Daan Krammer has constructed an analogue of the Tits representation for the cactus groups.
He proves this is faithful. The preprint version is:
</p>
<p>
<a class="ext-link" href="https://arxiv.org/pdf/0708.1273.pdf"><span class="icon"></span>https://arxiv.org/pdf/0708.1273.pdf</a>
</p>
<p>
[You need to leave out the six term relations.]
</p>
TicketBruceFri, 14 Jul 2017 14:48:58 GMT
https://trac.sagemath.org/ticket/19594#comment:36
https://trac.sagemath.org/ticket/19594#comment:36
<p>
Is there a class for permutation representations of a finitely presented group?
</p>
<p>
I implemented the action on highest weight words in the tensor power of a crystal many years ago
(using the Henriques-Kamnitzer definition).
</p>
TicketdarijSun, 16 Jul 2017 13:09:31 GMT
https://trac.sagemath.org/ticket/19594#comment:37
https://trac.sagemath.org/ticket/19594#comment:37
<p>
Bruce: Interesting! Can you direct me to the specific point where the representation is constructed? How do I "leave out" the six term relations? (I mean, what do I tweak to make sure that the representation is still faithful?)
</p>
TicketgitTue, 05 Dec 2017 14:32:46 GMTcommit changed
https://trac.sagemath.org/ticket/19594#comment:38
https://trac.sagemath.org/ticket/19594#comment:38
<ul>
<li><strong>commit</strong>
changed from <em>002edc4c3ba10274b93dbf960ed5a4d6d8e7f270</em> to <em>14ec15b7c35aecd879bc4bab020dddd19580d85c</em>
</li>
</ul>
<p>
Branch pushed to git repo; I updated commit sha1. New commits:
</p>
<table class="wiki">
<tr><td><a class="ext-link" href="https://git.sagemath.org/sage.git/commit/?id=14ec15b7c35aecd879bc4bab020dddd19580d85c"><span class="icon"></span>14ec15b</a></td><td><code>Merge branch 'public/groups/cactus_group-19594' in 8.1.rc4</code>
</td></tr></table>
TicketgitTue, 05 Dec 2017 19:06:12 GMTcommit changed
https://trac.sagemath.org/ticket/19594#comment:39
https://trac.sagemath.org/ticket/19594#comment:39
<ul>
<li><strong>commit</strong>
changed from <em>14ec15b7c35aecd879bc4bab020dddd19580d85c</em> to <em>b354f580ec5437e70b4d92502f09aa6206cb2fc1</em>
</li>
</ul>
<p>
Branch pushed to git repo; I updated commit sha1. New commits:
</p>
<table class="wiki">
<tr><td><a class="ext-link" href="https://git.sagemath.org/sage.git/commit/?id=b354f580ec5437e70b4d92502f09aa6206cb2fc1"><span class="icon"></span>b354f58</a></td><td><code>fix import</code>
</td></tr></table>
Ticketbsalisbury1Tue, 24 Jul 2018 14:16:53 GMTstatus, cc changed
https://trac.sagemath.org/ticket/19594#comment:40
https://trac.sagemath.org/ticket/19594#comment:40
<ul>
<li><strong>status</strong>
changed from <em>needs_review</em> to <em>needs_work</em>
</li>
<li><strong>cc</strong>
<em>bsalisbury1</em> added
</li>
</ul>
<p>
According to a conversation with Travis today, the mathematics of this patch is incorrect at the moment.
</p>
Ticket