Comments for Tom's Math Weblog http://trdunlap2.wordpress.com What I'm working on and what I'm finding Wed, 08 Jul 2009 18:50:18 +0000 http://wordpress.com/ hourly 1 Comment on GRTEALA 1: Review of the situation by trdunlap2 http://trdunlap2.wordpress.com/2009/07/08/grteala-1-review-of-the-situation/#comment-148 trdunlap2 Wed, 08 Jul 2009 18:50:18 +0000 http://trdunlap2.wordpress.com/2009/07/08/grteala-1-review-of-the-situation/#comment-148 I'll list the following with more detail in a future post For $latex L\mathfrak{sl}_2$: - Plucker relations are a wash so far. - My work has been assuming the Induction works - I believe reduction to dim-2 will still work - There will undoubtedly me infinitely many primitives - MV-polytopes won't uniquely factor into primitives without introduction of non-MV-polytopes: a sort of "imaginary" cluster. But Kamnitzer was surprised I would seek unique factorization anyway. - I'm excited to check the networks of non-overlapping cords. There is no proof (that I know of) of their correlation for finite type, only an observation. But would interest me if such a pattern would continue to hold in the affine case. I’ll list the following with more detail in a future post

For L\mathfrak{sl}_2:
- Plucker relations are a wash so far.
- My work has been assuming the Induction works
- I believe reduction to dim-2 will still work
- There will undoubtedly me infinitely many primitives
- MV-polytopes won’t uniquely factor into primitives without introduction of non-MV-polytopes: a sort of “imaginary” cluster. But Kamnitzer was surprised I would seek unique factorization anyway.
- I’m excited to check the networks of non-overlapping cords. There is no proof (that I know of) of their correlation for finite type, only an observation. But would interest me if such a pattern would continue to hold in the affine case.

]]> Comment on A conjecture on the Uniqueness of MV-Polytopes by trdunlap2 http://trdunlap2.wordpress.com/2009/07/06/a-conjecture-on-the-uniqueness-of-mv-polytopes/#comment-147 trdunlap2 Tue, 07 Jul 2009 22:43:09 +0000 http://trdunlap2.wordpress.com/?p=320#comment-147 The conjecture is obviously false as stated because I forgot another restriction. We need a kind of maximality. For example, in the case of $latex \mathfrak{sl}_2\times\mathfrak{sl}_2$ our polytopes may be rectangles (the "right" answer) or line segments -- or a number of other choices all of which consist of subsets. I also may have to assume some version of convexity (subject to condition 1) -- this I can't remember. The conjecture is obviously false as stated because I forgot another restriction. We need a kind of maximality. For example, in the case of \mathfrak{sl}_2\times\mathfrak{sl}_2 our polytopes may be rectangles (the “right” answer) or line segments — or a number of other choices all of which consist of subsets.

I also may have to assume some version of convexity (subject to condition 1) — this I can’t remember.

]]> Comment on Zero-stability: Examples by trdunlap2 http://trdunlap2.wordpress.com/2009/05/04/zero-stability-examples/#comment-146 trdunlap2 Mon, 04 May 2009 20:55:15 +0000 http://trdunlap2.wordpress.com/?p=316#comment-146 Note: I calculate $latex \mathfrak{z}^s_0$ in that last example to be $latex (\mathbb{C}^*)\times (\mathbb{C}^*)$. With sufficient choice of $latex \rho$ the defining $latex \mathbb{C}^*$-action will have four fixed points in the closure, none in the interior. Note: I calculate \mathfrak{z}^s_0 in that last example to be (\mathbb{C}^*)\times (\mathbb{C}^*). With sufficient choice of \rho the defining \mathbb{C}^*-action will have four fixed points in the closure, none in the interior.

]]> Comment on 0-stable Modules by trdunlap2 http://trdunlap2.wordpress.com/2009/04/24/0-stable-modules/#comment-144 trdunlap2 Fri, 24 Apr 2009 22:55:02 +0000 http://trdunlap2.wordpress.com/2009/04/24/0-stable-modules/#comment-144 Tomorrow I give an alternate definition of 0-stable and next week some example calculations. Tomorrow I give an alternate definition of 0-stable and next week some example calculations.

]]> Comment on Special case reflecting onto a vertical edge by trdunlap2 http://trdunlap2.wordpress.com/2008/12/18/special-case-reflecting-onto-a-vertical-edge/#comment-135 trdunlap2 Thu, 18 Dec 2008 08:19:59 +0000 http://trdunlap2.wordpress.com/?p=258#comment-135 I believe its only coincidental that the point of contention is the reflection of a reflection. I believe its only coincidental that the point of contention is the reflection of a reflection.

]]> Comment on Another example by trdunlap2 http://trdunlap2.wordpress.com/2008/12/17/another-example/#comment-134 trdunlap2 Wed, 17 Dec 2008 22:59:04 +0000 http://trdunlap2.wordpress.com/?p=254#comment-134 Next: Special case when corners reflect onto a vertical line. Next: Special case when corners reflect onto a vertical line.

]]> Comment on On the “longest element” by trdunlap2 http://trdunlap2.wordpress.com/2008/11/13/on-the-longest-element/#comment-129 trdunlap2 Sat, 15 Nov 2008 23:05:13 +0000 http://trdunlap2.wordpress.com/?p=200#comment-129 On second thought reflection through the origin might be more natural than rotation about the sl_2 axis. It produces the same set chamber weights because infact the two differ only by a reflection that's already in the Weyl group. On second thought reflection through the origin might be more natural than rotation about the sl_2 axis. It produces the same set chamber weights because infact the two differ only by a reflection that’s already in the Weyl group.

]]> Comment on Calculations by trdunlap2 http://trdunlap2.wordpress.com/2008/11/12/calculations-2/#comment-128 trdunlap2 Sat, 15 Nov 2008 23:03:30 +0000 http://trdunlap2.wordpress.com/?p=188#comment-128 I didn't mention, the dimensional results of my calculations are consistent with those given in "Loop Groups". I didn’t mention, the dimensional results of my calculations are consistent with those given in “Loop Groups”.

]]> Comment on Polytopes for LSL2 by Walls by trdunlap2 http://trdunlap2.wordpress.com/2008/10/14/polytopes-for-lsl2-by-walls/#comment-124 trdunlap2 Thu, 16 Oct 2008 06:07:54 +0000 http://trdunlap2.wordpress.com/?p=181#comment-124 Applying L'Hopital's rule to the each formula notice that the limit as either variable goes to plus or minus infinity is the other variable. Also note that $latex \phi(4,5)=\phi(3,5)=\phi(3,4)=2$ ($latex \phi$ here denoting the formula above.) So these three cells will never three meet. This can also be seen because their normal vectors lie on a line. Applying L’Hopital’s rule to the each formula notice that the limit as either variable goes to plus or minus infinity is the other variable.

Also note that \phi(4,5)=\phi(3,5)=\phi(3,4)=2 (\phi here denoting the formula above.) So these three cells will never three meet. This can also be seen because their normal vectors lie on a line.

]]> Comment on Polytopes for LSL2 by Walls by trdunlap2 http://trdunlap2.wordpress.com/2008/10/14/polytopes-for-lsl2-by-walls/#comment-123 trdunlap2 Wed, 15 Oct 2008 20:32:43 +0000 http://trdunlap2.wordpress.com/?p=181#comment-123 I'm not sure what placement of planes should be possible, but I'd guess that vertices should be on the lattice, which would pose a major restriction. I’m not sure what placement of planes should be possible, but I’d guess that vertices should be on the lattice, which would pose a major restriction.

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