GUT that includes all 3 particle families into a large group? Explaining SU(5) GUTs (using the first particle family as an example) in the last SUSY lecture 10, Lenny Susskind mentioned that there are at present no ideas how to combine simultaneously all 3 particle families into a large GUT theory.
I somehow dont believe him, suspecting that he just didnt want to talk about this :-P...
So, are there any ideas around how to incorporate all 3 families into a larger structer?
If so, I would appreciate explanations about how it works at a "Demystified" level :-)
 A: Jacob Bourjaily derives three Standard Model generations from an E8 singularity here and here. (See comments by Lubos, 1 2 3.) 
A: The problem of families in GUT is sometimes referred as an "Horizontal symmetry". There are two lines of work, roughly: those which get a continous symmetry, say SU(3), and then all the gauge malabars, and those which add a discrete symmetry, such as A4. Of course in both cases, a serious GUT should show everything embodied in a larger simple group. E8 has some value because it can go down to E6xSU(3), and E6 can lodge chiral fermions (but then perhaps this SU(3) does not work as it should, in more detailed examination) Other alternatives are just growing up SO(2n) until everything fits... You always have V+A currents you dont want, plus a bag of any of the usual problems in phenomenology.
Zee is the adecuate source to check if you want to look deeper in this topic.
A: There has been quite a lot of coverage about Garrett Lisi's Exceptionally Simple Theory of Everything. You can find the preprint here, as well as a SciAm special here.
Lisi's theory involves using $E_8$, which is the largest exceptional Lie algebra in Killing's classification. I'm not aware of the theory appearing in a peer-reviewed journal. I think Lisi claims to also incorporate gravity but, because the group is so large, there are also a lot of new particles that would need explaining away. The Wikipedia article is reasonably fair about the coverage of the original preprint and makes some attempts to explain it with pretty visualizations of $E_8$. The theory is highly controversial and doesn't have widespread acceptance. but it is an "idea around how to incorporate all 3 families into a larger structure"...
There are probably more conventional ideas that use SU(5) or SO(10), but they clearly don't have as good PR people because I'm not aware of any leading theory.
A: SO(8)'s triality can be used to generate three families. It's promising from a GUT perspective.
E6 and SO(18) are the best potential GUT groups containing SO(8) for doing so; of these E6 appears more "seamless". See the following:


*

*Z K Silagadze, SO(8) colour as possible origin of generations, arXiv:hep-ph/9411381v2 (2009).

*Y BenTov, A Zee, The origin of families and SO(18) grand unification, arXiv:1505.04312v2 [hep-th] (2016).

*H Rubenthaler, The (A_{2},G_{2}) duality in E_{6}, octonions and the triality principle, Transactions of the American Mathematical Society 360, No. 1, 347-367 (2008).

*M Ito et al, E_{6} grand unified theory with three generations from heterotic string, arXiv:1012.1690v2 [hep-ph] (2011).
