Awesome. Thank you! Out of curiosity is there a reason .norm_squared() doesn't just divide by 2 for us?
Joe On Tuesday, July 26, 2016 at 4:29:00 PM UTC-4, Travis Scrimshaw wrote: > > > > 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.
