Sorry for not responding earlier about this. I posted on the GH issue, but in short, you can use `W.domain().roots()` or passing it the explicit space that has the roots that you want to use, e.g.,
sage: W = WeylGroup(RootSystem(['G',2]).root_lattice()) sage: list(W) [ [1 0] [ 1 0] [-1 3] [-1 0] [-1 0] [ 1 -3] [-1 3] [ 2 -3] [0 1], [ 1 -1], [-1 2], [ 0 -1], [-1 1], [ 1 -2], [ 0 1], [ 1 -1], [-2 3] [ 1 -3] [-2 3] [ 2 -3] [-1 2], [ 0 -1], [-1 1], [ 1 -2] ] Of course, there is the way mentioned above using the reflection representation (in a "natural" basis and the Coxeter matrix data): sage: W = CoxeterGroup(['G',2], implementation='reflection') sage: list(W) [ [1 0] [-1 a] [ 1 0] [ 2 -a] [-1 a] [-2 a] [ 2 -a] [ 1 -a] [0 1], [ 0 1], [ a -1], [ a -1], [-a 2], [-a 2], [ a -2], [ a -2], [-2 a] [-1 0] [ 1 -a] [-1 0] [-a 1], [-a 1], [ 0 -1], [ 0 -1] ] Best, Travis On Monday, March 10, 2025 at 2:23:33 AM UTC+9 dmo...@deductivepress.ca wrote: > I have opened issue #39656 <https://github.com/sagemath/sage/issues/39656> > for further discussion, and expect to post a comment there soon, but > questions like this will probably get a better response from other forums. > > On Tuesday, March 4, 2025 at 5:42:38 PM UTC-7 dmo...@deductivepress.ca > wrote: > >> 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. 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