Since this is related to development, you can open an issue at https://github.com/sagemath/sage/issues. If another question comes up that isn't being answered, you could ask here on sage-devel, but the answer and discussion would continue on the issue site.
Anyway, I think an answer to your original question is provided by the representation_matrix of the reflection_representation, as illustrated by the following (but if there needs to be further discussion, then I think an issue should be opened): sage: G2 = WeylGroup(['G',2]) sage: [G2.reflection_representation().representation_matrix(g) for g in G2] [ [1 0] [ 1 0] [-1 a] [-1 0] [-1 0] [ 1 -a] [-1 a] [ 2 -a] [0 1], [ a -1], [-a 2], [ 0 -1], [-a 1], [ a -2], [ 0 1], [ a -1], [-2 a] [ 1 -a] [-2 a] [ 2 -a] [-a 2], [ 0 -1], [-a 1], [ a -2] ] On Tuesday, March 4, 2025 at 3:12:43 PM UTC-7 dim...@gmail.com wrote: > On Tue, Mar 4, 2025 at 11:48 AM dmo...@deductivepress.ca > <dmo...@deductivepress.ca> wrote: > > > > 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. > > Another option is sage-support. I added it to CC. > > Anyhow, I can give a quick answer. The choice of the representation > returned by WeylGroup() is allowing you to use matrices with rational > entries. > If you want to keep them rational, and keep the full group (i.e. have > a faithful representation), not its quotient, then there > is no way around it, you'd need to stay in the dimension given. > E.g. for G_2, which is isomorphic to the dihedral group of order 12, > there is no way to provide a faithful 2-dimensional representation, > you'd need to use certain irrational numbers. > > HTH > Dima > > > > > > > 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+...@googlegroups.com. > > To view this discussion visit > https://groups.google.com/d/msgid/sage-devel/177211ed-3b9d-4082-a7a4-dc33b7d10d76n%40googlegroups.com > . > -- 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/00534677-7c60-409b-a023-18bb90a3328dn%40googlegroups.com.
