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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