Clifford algebra calculations in galgebra
I performed extensive calculations using galgebra for a recent paper (as well as Sage). https://github.com/pygae/galgebra
However, the sample code of how to do these extensive calculations might be useful for other researchers and students in the Clifford algebra community, as I found tutorials/sample code a little limited. So it may be useful to see some extended Clifford projects in galgebra in action that people can see in action and adapt to their own purposes. Happy experimenting!
Thank you, Hugo Hadfield and Eric Wieser for help with setting up galgebra and Alex Konovalov for getting me started on github back when.
Arxiv link to paper https://arxiv.org/abs/2103.07817
Clifford spinors and root system induction: H4 and the Grand Antiprism
Recent work has shown that every 3D root system allows the construction of a correponding 4D root system via an `induction theorem'. In this paper, we look at the icosahedral case of H3→H4 in detail and perform the calculations explicitly. Clifford algebra is used to perform group theoretic calculations based on the versor theorem and the Cartan-Dieudonné theorem, giving a simple construction of the Pin and Spin covers. Using this connection with H3 via the induction theorem sheds light on geometric aspects of the H4 root system (the 600-cell) as well as other related polytopes and their symmetries, such as the famous Grand Antiprism and the snub 24-cell. The uniform construction of root systems from 3D and the uniform procedure of splitting root systems with respect to subrootsystems into separate invariant sets allows further systematic insight into the underlying geometry. All calculations are performed in the even subalgebra of Cl(3), including the construction of the Coxeter plane, which is used for visualising the complementary pairs of invariant polytopes, and are shared as supplementary computational work sheets. This approach therefore constitutes a more systematic and general way of performing calculations concerning groups, in particular reflection groups and root systems, in a Clifford algebraic framework.
galgebra: AACA_SpinH3.ipynb, Doubling-AACA.ipynb, Rotor_4D_mapping-AACA.ipynb
Sage: AACA_H4_GA_Sage.ipynb, Coxeter_Plane_functions.ipynb