Timeline for Are there positive formulae for the inner product between elements of a Lie algebra representation in the Shapovalov form?
Current License: CC BY-SA 2.5
11 events
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| Mar 3, 2010 at 21:11 | answer | added | David Hill | timeline score: 3 | |
| Mar 3, 2010 at 17:00 | answer | added | Peter Tingley | timeline score: 2 | |
| Feb 28, 2010 at 0:52 | history | edited | Ben Webster♦ | CC BY-SA 2.5 |
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| Feb 28, 2010 at 0:51 | comment | added | Ben Webster♦ | These are the coefficients of a Gram matrix for the Shapovalov form. There's such a matrix for any collection of vectors, and I think the one I'm looking at is not one people would usually choose (it's not a basis, just a spanning set). The Hill paper is interesting (thanks for pointing it out), but I don't think it really overlaps much with the kind of information I'm looking for here (in particular, he was most interesting in determinants, whereas I want actual coefficients). | |
| Feb 28, 2010 at 0:45 | history | edited | Ben Webster♦ | CC BY-SA 2.5 |
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| Feb 27, 2010 at 9:54 | comment | added | Vladimir Dotsenko | Just want to make sure I am not imagining things - are the inner products you describe are elements of the Gram matrix for the Shapovalov form? I think there are some elegant formulas for them in the affine case (there was some paper by David Hill a couple of years ago). Also, you might want to look at what they do when proving Kac formulas for Z(g)=S(h)^W where the Shapovalov form is used. | |
| Feb 27, 2010 at 5:06 | history | edited | Ben Webster♦ | CC BY-SA 2.5 |
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| Feb 27, 2010 at 5:03 | comment | added | Ben Webster♦ | Yes. I know a vector space that has that dimension. But it's not a vector space I have any hope of calculating a basis of and putting in bijection with something. | |
| Feb 27, 2010 at 4:23 | comment | added | Allen Knutson | Is there a reason to expect them to be positive? | |
| Feb 27, 2010 at 3:46 | comment | added | Theo Johnson-Freyd | Just to be clear, since [E_i,F_j] = \delta_{ij}H_i, the problem is just that F...Fv might be at a negative weight? Anyway, are you asking for a natural basis for the representation in which the bilinear form is given by a positive matrix? Then there is a (very not natural) answer: pick some very large number C, and let the basis consist of vectors of the form v + C^{-1} F...Fv, where v is your highest weight vector. If you'd rather have an integer matrix, let C be a very large integer, and take the basis to be things like Cv + F...Fv. | |
| Feb 27, 2010 at 1:04 | history | asked | Ben Webster♦ | CC BY-SA 2.5 |