I recently looked at Computing in Groups of Lie Type <http://www.win.tue.nl/~amc/pub/papers/cmt.pdf>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. 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.
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...) (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. (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." On Saturday, June 25, 2016 at 12:27:08 AM UTC-4, Travis Scrimshaw wrote: > > Hey David, > Thank you for the links. > > To add to all this: GAP has already implemented some related objects: >> (1) Lie algebras >> (http://www.gap-system.org/Manuals/doc/ref/chap64.html, >> http://www.gap-system.org/Datalib/lie.html, >> http://www.science.unitn.it/~degraaf/sla.html >> <http://www.google.com/url?q=http%3A%2F%2Fwww.science.unitn.it%2F~degraaf%2Fsla.html&sa=D&sntz=1&usg=AFQjCNHWNpq02QOvvyRIlenKNiwUjFz5pg>) >> >> >> > > Implementing an interface with GAP's Lie algebras and related objects is > something on my todo list as well. When I started #14901, I was not as > aware of GAP's capabilities, and I think we have much better linear algebra > algorithms that we can leverage. Although my current implementation doesn't > do a very good job of this... > > I need to implement an optional spkg for QuaGroup and an interface to it. > I guess this should also include SLA. A question is should the GAP packages > be bundled together. I will start another thread about this. > > 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.
