Estimated Submission: Spring Break 2009 (finish write up last 5%, troll out and edit manuscript, lastly smooth over rough spots)

Submitted: awaiting response

Re-read entire paper againFollow link Fix rank three notationsFollow link Add in several more references on diagramsFollow link Add preliminaries on diagrams, and basic examplesFollow link Respond to Sean's editsFollow link Add discussion of signsFollow link Explicit barbell computationFollow link write diagramatically the rank 3 relationFollow link |

(1) add in section 1 editsFollow link (2) add in section 2 editsFollow link (3) add in section 6 editsFollow link (4) make jarnal file edits on sections 3-5 (and where to add in diagrams elsewhere)Follow link edit entire article (Monday 8/4)Follow link discuss possilbe algorithms to compute reynolds graphically (decided to leave as comment only in this paper and possibly explore further at a later time)Follow link Update with Jarnal editsFollow link Add in Reynolds commentFollow link add in shur like lemma commentFollow link Figure out relevance to Reynolds OperatorFollow link add more documentation to public mathematica file (LOW PRIORITY)Follow link update Wolfram library (LOW PRIORITY)Follow link make note that program seems to not run appropriately in Mathematica 7 (better yet try to debug for mathematica 7)[low priority]Follow link Prove (via spin networks) shur like lemma (CUT FOR NOW)Follow link compute the reynolds operator in terms of spin networks in some examples (CUT FOR NOW)Follow link |

Point Distribution Algorithm (edited by elishapeterson on 01 Mar 2010 18:41)

Trace Diagrams Collaboration Page (edited by Sean Lawton on 30 Nov 2009 15:21)

rank3.working.wiki.tex (edited by elishapeterson on 09 Feb 2009 19:01)

nonsimplerecurrence.tex (edited by elishapeterson on 09 Feb 2009 19:01)

simplerecurrence.tex (edited by elishapeterson on 09 Feb 2009 19:00)

tracediagramreview.tex (edited by elishapeterson on 09 Feb 2009 19:00)

rankrcentral.workingwiki.tex (edited by elishapeterson on 09 Feb 2009 18:59)

spinnetsintro.tex (edited by elishapeterson on 09 Feb 2009 18:59)

intro.workingwiki.tex (edited by elishapeterson on 09 Feb 2009 18:57)

spinsl2-ii.workingwiki.tex (edited by elishapeterson on 09 Feb 2009 18:56)

Collaboration Pages (edited by elishapeterson on 07 Jul 2008 23:12)

## TeX Links (working wiki pages)

- spinsl2-ii.workingwiki.tex
- intro.workingwiki.tex
- spinnetsintro.tex
- rankrcentral.workingwiki.tex
- tracediagramreview.tex
- simplerecurrence.tex
- nonsimplerecurrence.tex
- rank3.workingwiki.tex

Supporting Files

## Announcements

- Sean is working on at present.
- Elisha is working on at present.

## Publication Information

- First suggestion:
*Geometriae Dedicata*(to Wentworth or García-Prada)

## Outline (copied from PDF)

Contents

1. Introduction 2

2. Rank r Central Functions 2

3. Decompositions and Central Functions 2

4. Review of Trace Diagram Central Function Representation 2

4.1. Notation 2

4.2. Fundamental Spin Network Identities 2

4.3. Modifying Spin Network Identities 3

5. Simple Loop Recurrences 4

5.1. An Essential Lemma 4

5.2. The Main Result 5

5.3. Left Associative Central Function Recurrences 6

6. Non-Simple Loop Recurrences 8

6.1. A General Recurrence Formula 9

6.2. Barbell Recurrences 10

7. Rank Three Central Functions 11

7.1. Tensorial Algorithm 11

7.2. Combinatorial Algorithm 14

7.3. Computations 15

7.4. Shur Like Lemma 17

8. Symmetry and Central Functions 17

References 17

## References

Elisha's Thesis

"Spin Networks and Character Varieties" paper

PGF/TikZ Manual (large!)

Goldman Paper

I have edited and changed the sections I wrote. I also uploaded a new pdf file called spinsl2-ii.workingwiki.pdf. I am now going to create a Jarnal file with some initial comments on the sections you wrote. Please do the same, if relevant, on the section I wrote. If you have edits on your sections, upload them to the site too. Talk to you soon.

ReplyOptionsI just finished looking over the paper again. Overall, I like how it's coming together. I still think the mixture of the algebraic approach and the diagrammatic approach is good.

I was thinking of trying to include a lot more of the diagrams with the algebra and the generic construction of central functions. I actually think this might help the reader to see how it works, especially when we're describing how a general tensor product decomposes (bottom of page 6). It might be nice to parallel the algebraic construction with the diagrammatic construction.

Also, I would like to shift my material slightly to focus more on the algorithmic rank 3 case. I can briefly discuss the work involved in directly evaluating the diagrams (which produced my first set of results)… fairly straightforward to implement but computationally hard. Then I could cover my recurrence algorithm in the rank 3 case, and provide the mathematica files online to go along with it. I'd then briefly indicate how the process could be extended to a more general case.

The last thing is that I'd like to try to emphasize the importance of central functions a bit more in the paper, beyond citing our previous work with them. Do we know of any interesting properties of central functions? I think that might help to engage the reader. Perhaps we can explore the connections with invariant theory a bit more. I think there's some interesting stuff there.

ReplyOptionsI agree that there is some interesting relationships between the spin networks and invariant theory. Perhaps we can take a look at how the rank 3 relation (there is only one) which generates the ideal looks like in terms of diagrams. This would be interesting. In general, every coordinate ring decomposes as a direct sum with its ring of invariants (using the so called Reynolds operator). This decompostion is NOT a ring theoretic direct sum, but ONLY a G-module direct sum decompostion. I suppose the central functions are related to the Reynold operator. This could be interesting, since typically the Reynolds operator is used in invariant theory in an existential fashion, but not a constructive one. I guess our work is related to the constructive invariant theory of Reynold's operators. I will have to think further about this. In any event, I am starting to believe that graphical methods may be important in the relations problem. Working out the example in this case should be interesting enough and engaging.

And I agree on both points: 1. adding diagrams throughout is a good idea and 2. focusing more on the rank 3 case is a good idea. However, I think one of the important points of the paper is that iterating the rank 2 constructions allows for general rank r algorithms; so we should indicate how and at least offer the reader the correct set-up even if we only follow through with it for the rank 3 computations (I think we have more or less done this).

ReplyOptionsI updated the rank 3 file with a brief start and an outline, and copied the tex back to the wiki.

Would you prefer to update the page with the tex code, or just the file? I'm leaning to updating the tex code, because that will keep track of all changes made to the page. Perhaps that would make it unnecessary to even keep the .tex files below?

Also, I'll get my comments to you on the intro and first section tomorrow via jarnal.

ReplyOptionsLets keep the wiki page updated and cut and past "dated files" from there on our hard drive as back-ups (this is of course optional, but I like doing this every few days during active updating).

ReplyOptionswe should keep an updated PDF as a link somewhere however on a daily basis . also in the wiki version, in the code, we should ONLY have TeX; that is no wiki mark-ups. This makes it very easy to cut and paste into local files.

Okay. I added a new section to this page which list the most recent edits to files, so we can keep track of what was last changed.

ReplyOptionsI found one of the issues I was having with the recurrences. There's a small typo in our previous paper that changes the sign of one of the 6j-symbols. I've fixed that and now I'm working to see if that helps my code.

ReplyOptionsMake a comment in this new paper correcting the mistake, since the published version of our first paper will have the typo.

ReplyOptionsI have written up (in separate files) a revised version of the presentation of the simplerecurrence/nonsimplerecurrence portion of the paper. I am fairly happy with how it has gone. Before I integrate it back into the paper, I want to work out (& test) explicitly the "barbell" portion of the recurrence formulas. I am going to make the diagrammatic theory much more directed towards the rank three combinatorial formula. So hopefully sometime this week I'll get that updated.

ReplyOptionsGreat. Once you update it, I can take a look and start to edit again. Take a look at my updates and let me know if you approve or not.

I am going to try to prove the "shur like lemma" with spin networks and also to write out the rank 3 relation via spin networks. I may need your help however.

Once we have a good draft, if you want we can talk about my Reynold's operator comment in the paper (but this should be a lower priority).

ReplyOptionsThis now my number 1 priority (for work). I will be moving for the next 3 weeks and about August 10th, I will have to start to plan my courses for the Fall. But I should be able to focus on this (off and on) for the next 10 days (at least). We need to talk about "next steps". Let me know what a good time to chat with you is.

ReplyOptionsI suggest we change the title to

Computing SL(2,C) Central Functions with Spin Networks

Thoughts???

ReplyOptionsTeX Links 1, 2, 3, and 7 are updated.

Everywhere you said you want to add in diagrams, I thought was a good idea. I addressed all your comments (they were good, thanks!). I wrote %diagrams here at the places you said you want to add diagrams…have at it.

ReplyOptionsI just added the algebraic form of the product formula….so it is done.

ReplyOptionsI updated the abstract. I think it is an improvement, but I am not sure I like it. Anyway, feel free to improve it if an improvement come to mind.

ReplyOptions