On Tuesday, July 26, 2016 at 3:07:04 PM UTC-5, Joseph Hundley wrote: > > I recently looked at Computing in Groups of Lie Type > <http://www.google.com/url?q=http%3A%2F%2Fwww.win.tue.nl%2F~amc%2Fpub%2Fpapers%2Fcmt.pdf&sa=D&sntz=1&usg=AFQjCNEK0V9M1i2uGHXlVWVU-HgDTokI9w>by > > Cohen, Murray and Taylor. The basic approach to representation and > calculation which is taken there is essentially the one I had in mind. Plus > they've worked a number of details I thought I would have to work out. It > appears that this article was written as the algorithms in question were > being implemented in Magma. I wonder firstly whether it would be considered > "kosher" to reimplement them in Sage, and secondly whether it would be > considered desirable. >
Yes and possibly yes. It depends on how much of the algorithms depends on doing linear algebra (or other aspects of Sage). If the answer is very little to none, I would do it as a separate package in C++ for the speed and then construct an iterface to it using Sage. Although we can always do it in Sage and extract things out afterwards. > Naively, it seems to me that the article would not have been written if > the algorithms were to be treated as protected intellectual property. And > I'd already stumbled across quite a bit of it independently by following my > nose. Still I'm fairly new to programming and have not yet learned the > ethos around such things. > Its a published research paper, so we (i.e., anyone) are free to provide our own implementation. > > If I were going forward with this, I think a reasonable first step would > be to equip root systems with the ability to compute a few key integral > invariants of roots and pairs of roots. Including: > (1) the (normalized integral) norm square (always 1 for short roots, 2 or > three for long...) > This is essentially already in Sage; you just need to divide by 2: sage: al = RootSystem(['B',4]).root_lattice().simple_roots() sage: al[1].norm_squared() 4 sage: al[4].norm_squared() 2 sage: al = RootSystem(['G',2]).root_lattice().simple_roots() sage: al[1].norm_squared() 2 sage: al[2].norm_squared() 6 > (2) the largest integer k (given roots a and b) such that a+kb is a root. > (3) the smallest integer k (given roots a and b) such that a+kb is a root. > I've done these ad-hoc for the classical Lie algebras (see #16821), but it would be good to have these as general methods. > (4) integral structure constants of the Chevalley presentation. These are > not uniquely determined by the system, but there is a fairly standard way > of selecting them-- described for example in Cohen Murray Taylor or a 1988 > article of Gilkey and Seitz-- which depends only on the system, and > therefore seems to me like a method which should live in class "root > system." > If I am understanding what you want, see the structure coefficients for the classical groups in #16821. We can probably extract that out as separate functions, but I would not have them be a method of a root system since it is specialized for Lie algebras and groups of Lie type. We might be better at this point moving the discussion to a trac ticket. Best, Travis -- You received this message because you are subscribed to the Google Groups "sage-devel" group. To unsubscribe from this group and stop receiving emails from it, send an email to sage-devel+unsubscr...@googlegroups.com. To post to this group, send email to sage-devel@googlegroups.com. Visit this group at https://groups.google.com/group/sage-devel. For more options, visit https://groups.google.com/d/optout.
