I continued to play around with my BibTeX style files today. I had been using the style plainurl, because it supported hyperlinks in the bibliography to eprints on the arXiv, but other than the hyperlinks, I wasn’t very happy with the way it formatted things. So today I finally got around to running the Perl script urlbst on my amsplain style file. The upshot is that my bibliography now has AMS style formatting and hyperlinks to eprints on the arXiv. The downside, if you want to call it that, is that the references in my bibliography are now numbered in the form [1], [2], [3], etc., instead of 1., 2., 3., etc. There’s probably a way to change it back to the non-bracketed numbering style, though these BibTeX style files are so unreadable that I’ll be darned if I can figure it out.
Something else I figured out how to do: Suppress the Mathematical Reviews information that the AMS BibTeX style files append to the end of each bibliographic entry. Just comment out the line “mrnumber output.nonempty.mrnumber” in the function fin.entry.original in the AMS BibTeX style file. But as soon as I figured that out, I started to wonder why the AMS style files don’t just automatically create hyperlinks in the bibliography to the relevant page on MathSciNet. It looks like some of the code to make that happen is already there in the style file. Maybe there are customized style files for certain AMS publications that already create hyperlinks to MathSciNet? If the Mathematical Reviews information is going to appear in my bibliography anyway, I’d just assume it be a clickable hyperlink when I read the paper as a pdf.
. Here 
is the first Frobenius kernel of the unipotent radical of a Borel subgroup of a simple, simply-connected algebraic group 
over the field 
. The field 
, with 
of the root system 
of 
and 
. This is a borderline case for type 
, and our conjecture for the structure of 
in type 
, the region for which we already have a proof in hand. (In type 
.)
. The calculations I wanted to do get messier as the rank of the root system increases (
, while 
), and that’s why I was using GAP.
, and given an element 
of 
are elements in the Weyl group 
of an indecomposable root system 
is a sum of positive roots, and 
, where 
is one half the sum of all the positive roots in 
is a sum of distinct positive roots in 
. I wanted to know if Equation 
. First I looked at the case when the root system 
. There 
and 
. In retrospect, it was a bit silly to look at that example, because if 
. So of course I found no solutions with 
and 
be a set of simple roots for 
long and 
short. Then 
and 
. Let 
and 
be the corresponding simple reflections. These two simple reflections generate 
, and we have
, let 
(the longest element of 
, the identity element in 
, 
, and 
. Then
of weights. If you don’t know what a root system is, go read The Unapologetic Mathematician’s 
of positive roots, and the set 
of negative roots. If 
, is a negative root. The weight lattice 
.
be the subset of 
that can be written in the special form 
, with 
a sum of distinct positive roots, and 
a sum of distinct negative roots. If you are familiar with the theory of Lie algebras, then 
, the exterior algebra on the simple, complex Lie algebra 
associated to the root system 
, then 
implies 
.
and 
be distinct elements of 
equal to 
?” This is a generalization of the first question, since we could have 
.
, then two distinct elements of 
. I will describe below one solution (out of several) found by my program.
be a set of simple roots for 
, the sum of all positive roots in 
, the sum of the negative roots 
, 
, and 
. Then 
.
-substitution 
and 
:
, which is not the correct answer. For the same reason, the second and third expressions in the above equation are not equal. We talked about how you could change the bounds of integration, and then not need to undo the substitution, or about how you could choose to do the indefinite integral separately instead.
. Equivalently, we’re studying the cohomology ring for the restricted enveloping algebra of a certain nilpotent Lie algebra 
(the nilradical of the Borel subalgebra of the Lie algebra of 
, the tensor product of a symmetric algebra (polynomial ring) with the ordinary Lie algebra cohomology of 
, assuming 
is greater than the Coxeter number 
. We ran into some trouble trying to lower the bound to 
, but I had some ideas last night that might enable us to do it. The tricky part is that my idea involves getting our hands dirty with explicit cocycle representatives, and arguing that certain coproducts are zero by explicitly showing that certain maps are coboundaries. It could get messy.
