Doing Without Tropical Plücker?

Reading a more recent paper by Kamnitzer (arXiv:math.QA/0505398), he mentions a conjecture by Anderson-Miković which would inductively construct MV-polytopes without reference to the tropical Plücker relations. The conjecture is not true in general, but is true for for example. It may be something to look into for since none of the relations listed in Kamnitzer’s first paper apply and I don’t yet understand the mechanism by which they arise.

Update and Items for Feb14-Feb16

Sorry, I’ve been in bed for about a week trying to recover from some bug. I finally got back to work on Wednesday and I’m moving back in the swing of things. I don’t really feel totally recovered to be honest, but I can’t take any more time off. I certainly have been quiet on this blog for too long.

Today I drew out the labels on the affine plane of (co?)roots of . My goal is to work out the calculations like I did for and give them geometric interpretations. My next step is to find representative vectors from each weight space .

Yesterday I dug up the Atiyah paper where I saw the parabola mentioned in my last post. But it doesn’t seem to be quite what I’m looking for, at least I don’t recognize it yet. So I’m trying to beat my way through Loop groups chapter 9 where there’s also a parabola. I need to find my notes from last year if any still exist.

Items for Feb3-Feb6

DN wants me to explain the parabola diagram of for , the (co?)weight lattice I guess. Probably a longer discussion beginning with the weight diagrams for .

I’m still unpacking Loop groups. At least I should finish chapter 5.

To balance this algebra stuff, I’ll be thinking about appropriate valuations on (the “loop loop group”) and the combinatorial/geometric implications – or whatever that means.

Items for Jan 29-Jan 31

I will do some calculations described in 5.2 of Loop Groups. They say for example that their generators for can easily be shown to satisfy the relations which define . But without giving it dedicated scrutiny I don’t even know how the calculations look.

I will try to rediscover my understanding of the proof that is a semidirect product and explain it here as soon as I do.

I will peruse chapter 4. Perhaps I can find the necessary material to press on to Chapter 6 next week.

Items for Jan16-Jan19

I was reading chapter 9 of Loop Groups but now have moved back to chapter 5. I had skipped over it trying to get to the representations of but chapter 5 is where they discuss the root system in .

I really enjoyed the talks I went to at U Chicago last week. Besides categorification and quantum groups which both seem fascinating, toward the end the speaker did touch on but in some sense more subtle a notion than . Something about “homotopy fiber product” or something like that with regard to a diagonal map. I wish I could post the commutative diagram. Anyway if I do get distracted this week by those things I’d rather get distracted about the material rather than the first two.

Items for Jan10-Jan13

I did make some calculations regarding towers, but the subject remains open. I’ll try to post some more pictures next week.

For the remainder of this week I’m reading about representations of Loop groups from Pressley & Segal’s book. I’ve skimmed over chapter 9 already. But I need to work out exercises.

Items for Jan6-Jan9

I’m back from break. I had a meeting with DN yesterday. Here’s a list of things he gave me to keep in mind:

1. Plucker Relations
2. The Moment Map
3. Representations of Loop Groups

Over the next few days I want to look into:

• Making diagrams with latex in them. The only shot of this right now will have to be with xfig.
• Investigating the towers. In particular what the “middle” level values describe.
• Loop Groups

Items for Dec 9 – Dec12

• Posting my calculations.
  • Along those lines: Making pictures for upload.
    • I can use GIMP or POV-Ray as I’ve done before,
    • but I should learn to use some program that will output to Tex or a Tex-friendly format. (Suggestions?)
• I still have to figure out the polytope-cycle relation. Work out an example or two.
  Maybe:
  • start with a Weyl polytope,
  • then some trivial
    pseudo-Weyl polytope,
  • then an MV Polytope.
• Blog maintenance:
  explaining what’s going on, making it understandable to someone besides myself.