- if you can do it, fine, but since I believe the extra Z/2 corresponds to an observed antiunitary symmetry of the Standard Model (namely CPT symmetry), I'm going to take it as a win that the elegant stabilizer includes this extra Z/2.
We'll learn more when we figure out the best way to get not just the Standard Model gauge group but also its representation on fermions. And this is what I'm working on now, with Latham Boyle and his student Endre Bokor!
There are some interesting twists and turns here, but we're hoping to get a paper out soon.
