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/774b021e-c33f-4185-8c84-80224f7bba71n%40googlegroups.com.
