Who knew that my love of special relativity would collide with my love of data analysis? In the ongoing conversation between Soledad Villar (JHU), Ben Blum-Smith (NYU), and myself about writing down universally approximating functions that are equivariant with respect to fundamental physics symmetries, a problem came up related to the orientation of sets of vectors: In what groups are there possible actions on d-dimensional vectors such that you can leave all but one of the d vectors unchanged, and change only the dth? It turns out that this is an important question. For example, in 3-space, the orthogonal group O(3) permits this but the rotation group SO(3) does not! This weekend, I showed that the Lorentz group permits this. I showed it constructively.
If you care, my notes are here. It helped me understand some things about the distinction between covariant and contravariant vectors. This project has been fun, because I have used this data-analysis project to learn some new physics, and my physics knowledge to inform a data analysis framework.