garrett

MTd2:
Thanks for the advice. I like posting at PF because I feel I can speak more candidly here, and I like the checks and balances that are in place to keep the community polite. Hexality is just the product of duality and triality, so I'm not sure it really needs its own name. How to get three generations is still an open question.

Coin:
Distler's use of chirality is nonstandard -- it usually refers to how the weak force interacts with only left chiral fermions -- so I prefer to speak of complex and non-complex representations, which I think is what Distler is calling chiral and non-chiral. But, as I've explained above, it is possible to find a "chiral" representation space as a subspace of a "non-chiral one." The standard model algebra I'm working with is the usual algebra of the gravitational so(1,3) and standard model s(u(2)xu(3)) acting on the 64 dimensional representation space of one generation of fermions. This is the algebra that I find embedded in E8.

What I do next is look at how a triality automorphism maps this algebra into other parts of E8, and see if I can figure out how to relate this to the other two fermion generations. It is Distler's red herring to look directly at how the rest of E8 transforms under the gravitational and standard model subalgebras, as if that is what I'm doing, because I'm not. What I'm doing is to embed one generation of the standard model and gravity in E8 and then see where I can go from there. To say that there's a conjugate represenation space to the generation of fermions in E8 that makes that generation "not there" is silly in that context -- it depends on false assumptions about how E8 is being broken up and interpreted in my work.