This is a question about how to use sagemath, not an issue about sagemath development, so you should post it on a different forum, such as ask.sagemath.org. The people there are very knowledgeable.
On Tuesday, March 4, 2025 at 8:06:09 AM UTC-7 aska...@gmail.com wrote: > Hi, > > This is a long shot as it may be mathematically quite technical, but I > though I'd give it a try here. > > Sage has a library for Weyl groups associated to root systems: > > https://doc.sagemath.org/html/en/reference/combinat/sage/combinat/root_system/weyl_group.html > > A Weyl group is associated to a root system of type Xn where X is of type > A, B, C, D, E, F, G and n is a natural number, which is restricted > depending on the type. I won't explain more, as if you haven't seen this > before, an email is unlikely to be the best way to find it. > > The point here is the root system can be embedded in a lattice within RR^n > but also sometimes it is convenient to embed it in higher dimensions (e.g. > to avoid non-rational lattices). However, for my own purposes I really want > it embedded in RR^n, as I want the elements of the Weyl group (which is > generated by reflections through the hyperplanes orthogonal to at least one > of the roots) to be elements of GL(n, QQ[sqrt(d)]) for some integer d. If > the root system is embedded in a higher RR^m, so is the Weyl group. When > the root system is special (E6, E7, E8, F_4, G_2) this is possible and I am > sure it has its merits, but for good reasons, I actually need them embedded > in RR^n. > > For instance, it seems that the implementation in Sage has the root system > G2 embedded in RR^3 (so the matrices of WeylGroup are 3x3 matrices instead > of 2x2) and E6 and E7 are embedded in RR^8 and not RR^6 and RR^7. An easy > way to see this is to run > G= WeylGroup(['G',2]) > for g in G: > print(g) > and see how you get a list of 3x3 matrices and not 2x2. > (although you probably want to stop it before it finishes as there are too > many of them). > > My question is: is there any way, for these 3 root systems (G2, E6 and E7) > in Sage to have them embedded in the lowest possible lattice so that I get > the matrices of the Weyl group represented as matrices of size 2, 6 and 7, > respectively? For instance for G2, I know, mathematically, I could find a > projection from the lattice in RR^3 to RR^2 and use this projection to > project the matrices. I don't want to reinvent the wheel, though, so if > this is already existing in Sage that would be great. > > To be clear, I don't care about the other groups other than G2, E6, E7, > since for them I have already arranged it differently. > > Apologies if some of the above is not precise enough, I have a very > passing knowledge in representation theory. > > Many thanks in advance. > > Jesús > > > -- 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 view this discussion visit https://groups.google.com/d/msgid/sage-devel/177211ed-3b9d-4082-a7a4-dc33b7d10d76n%40googlegroups.com.
